The villain doesn't need to convince the protagonist as much as

you need to convince the reader.

If you can make it believeable to the reader that the protagonist changes sides, then it will be a satisfying read.

We often find out that our suspicions were mistaken in real life. For example, quite often a law enforcement agency finds out that their suspect was innocent. Why shouldn't that happen in a novel?

But you might also show how the villain manipulates the protagonist succesfully into believing an untruth. Protagonists don't always succeed, and this might be a story about a protagonist succumbing to some flaw. This too is something that happens in real life, where we often like to believe what we think should be true, but in fact isn't. If you manage to narrate this process convincingly, the reader will gladly follow you to your unhappy ending.

It's an unhappy ending if good does not triumph over evil.

Books and movies with unhappy endings are generally frowned upon, they tend to do poorly commercially. The successes amongst unhappy endings tend to be highly emotional, understandable cautionary tales; which means in a twisted way good still triumphs over evil, or at least evil leads somebody to a terrible end. The reader is following along with a character they like that descends into drug addiction, or crime, or losing their marriage, or corruption, etc, but they end up miserable or dead in the process.

The Escape Hatch ... Flip The Script

AKA role reversal. The villain proves to the MC that the MC has been duped, that he is a pawn of the people he thinks he is helping, or his management, etc. The villain doesn't convince the MC to be evil, the villain convinces the MC he has been working toward evil ends and the villain is doing good.

And instead of insisting the MC change sides, the villain gives him a chance to verify all this himself. After telling him all this, the villain, with the MC bound and a knife to his neck, says "I could kill you, and eliminate a threat. Remember that. But I won't, because I don't think you are evil. I'm going to set you free. Come back when you're ready to fight for what's right."

The reader following the MC is not disappointed, the twist does not mean the MC is now fighting on the side of evil. It means that the reader, like the MC, was duped into believing the villain was evil, but now understands the villain was good and the real villain is the queen that sent the MC out in the first place, trying to get rid of him so she can kill her husband and take power.

But the villain didn't kill the MC as she expected, the villain just pulled back the curtain to expose the queen as the real villain all along, and now the reader can believe the MC is still fighting for good, and the defeat of the queen represents the triumph of good over evil, so this is a happy ending.

Another Escape Hatch: Double Reversal.

A double reverse means the protagonist succumbs to the temptations of evil, but the "good" inside him eventually reasserts itself, and he reverse again: To the side of good, and defeats the villain after all.

Falling for the villain's talk is just another obstacle and failure along the way, this time within himself. You have a flawed protagonist. The next story is not exactly your situation, but consider a good cop. In a time of incredible financial hardship (e.g. he can't pay for the treatment his wife/mother/kid needs) he becomes a dirty cop. He gets deeper and deeper into being a dirty cop, until he accidentally shoots and kills an innocent witness he was trying to keep from exposing him. He covers that up successfully, but it weighs on him so much, he sacrifices himself in a blaze of glory to become, once again, a good cop.

Evil Triumphs In The End is not generally a story people like. It is depressing, and we read fiction to escape the real world, where evil often does triumph in the end, criminals and murderers and rapists and frauds get away with their predations, get insanely wealthy and/or powerful, and are never punished, dying peacefully in their sleep without a regret in the world.

You can write it if you want, perhaps it will be some kind of catharsis, but if your goal is to entertain people and have them like your story, then I'd suggest evil can be wildly successful in your story, but in the end, good must triumph over evil.

It could be interesting. Not all villains seem inclined to the talk, but having the hero listen and realize he was wrong could be refreshing.

Assuming your villain is a three dimensional character and your MC is likewise fully fleshed, there is no reason for this to feel a cheat. It would be a twist and likely change the direction of your story, but that can be a good thing.

Making a change of world view a logical move will be delicate and intriguing. What might induce the hero hell bent on destroying the villian to have an epiphany?

In one work I have set aside for the time, I was well into it when I discovered that my hero was actually an unknowing villain and the person he feared and hated the most was actually more of a reluctant hero. If my original ‘hero’ were not so blinded by his own prejudice, he could be persuaded that he was mistaken regarding the other man.

I mention that because your character must be capable of listening and hearing the truth even when spoken by the alleged villain.

I'm adding my two cents to the answer of Rasdashan:

It can be refreshing, but you have to do it well

The main issue is that such a unexpected change has to be foreshadowed. If the story progressed exactly as you described, it would feel awful for the reader.

Imagine the classic setup:

Hero goes to villain -> gets captured -> Villain does villain talk -> Hero has change of heart

The first three steps are a well known, widely used tropes. If you add the last step without any warning signals, it will be too sudden and readers will be dissatisfied. In other words, you can't do it just for the sake of subverting a trope; you have to justify it and foreshadow it.

Make the whole contrast between good and evil be less black and white, and more in a moral grey area, where the villain is surely questionable, but the good guys are also.
Make the hero skeptical about some things happening on his side. Make the villain convincing, and give him some pretty good proofs that he's not doing evil per se, but his evil acts are the result of necessary sacrifices in a conflict.

A fervent paladin of light won't turn to evil over a talk. The seeds for his turning, and hence the plot twist, must be planted way before the actual twist. Your hero must stop and consider his actions, the actions of his enemy, and doubt the very nature of the conflict. Make the villain clear his doubts. Make the villain answer some of those pesky questions. Make the villain show "good faith", e.g. releasing the hero, making him see firsthand that his armies are, maybe, just badly portrayed, and that nobody in his city is drinking mulled wine from the skulls of young children.

Adding more on foreshadowing, feel free to see this question: Writing SE
I remember Brandon Sanderson saying, in an episode of the first season of the podcast writing excuses, that twists must be foreshadowed at least thrice. So, consider that.

TL,DR: place your conflict in a moral grey area. Foreshadow.

The villain doesn't have to convince only the hero. The readers must be convinced too, or at least convinced that the whole situation is believable.

A Number of Choices

Villain is Not Truly Evil

In spite of our initial impression, the villain is either: doing something good, completely misunderstood, or falsely portrayed to the protagonist.

Even if he is planning something and upsetting some people, it may not end up hurting anyone (not anyone who matters, at least).

The villain could even be trying to help. Imagine if I said I wanted to infect a bunch of people with cowpox (mild illness) first, and then inject them with smallpox (severe illness) afterward. That sounds horrible unless you know that I am proving a theory of vaccination which suggests they will become immune to smallpox after recovering from cowpox.

Hero Becomes Evil

This is usually a punch to the gut if the reader actually likes the hero. Maybe it is temporary. However, if this change is permanent then your entire story has to be built to accommodate it.

Betrayal

The hero is only pretending to change sides. By the end, the villain will be confronted in satisfying fashion.

If you intend to keep this a secret, you'll have to choose carefully which type of narrator you use; readers will feel cheated if you have a close narrator who knows everything the hero thinks but "hides" this plan.

You'll need to provide some credible reason for the villain to accept the change of heart.

Means to an End

The hero has realized something is at stake which is far more important than the current conflict. The hero teams up with the villain to deal with it---perhaps sincerely, or perhaps with the intention of disposing of the villain along the way.

You will have to tie the villain's plans or resources into the larger issue somehow, and readers must believe that the players and their decisions are authentic.

This is the main plot point behind the Anime series, Maoyu. The "Hero" goes to fight the "Demon King" to end the war and suffering of all the human people, but ends up being turned to the Demon King's side.

The Anime accomplishes this in a few ways:

• 
  

  It doesn't give us much time at all to feel the Demon King is actually a terrible evil.

  
  
  

      The Anime opens with the Hero about to enter the Demon King's castle to face off. We're only really told this is a great evil force, not shown. This makes it easier to accept that maybe the Demon King isn't so evil after all.

  
  


• 
  

  It makes the Demon King completely opposite of every expectation both the Hero and viewer have.

  
  
  

      This is what really gets the Hero to actually listen in the first place. Perhaps your Ultimate Evil is actually a 14 year old, trying their best to fill the shoes of those before them while slowly changing the evil organization.

  
  



• It takes pains to identify the "true evil" (and show this time, instead of just tell) and dissociates the Demon King with it.

These are the three biggest methods I see to defy the viewer's expectations and turn what might be hate into empathy. How you accomplish this is up to you, but your end goal shouldn't be convincing the Hero as much as convincing the the reader. If you can't win the reader over to the other side, they'll be stuck hating your main character, no matter how realistic it is for the MC to act this way.

Most of the answers seem to centre around the revelation that the villain is not quite so evil as he was first made out to be and the protagonist isn't really "turning evil" so much as either realising he was mistaken or turning out to be good after all. This isn't really in the spirit of the question, IMHO.

You may be looking for an answer where the protagonist does indeed genuinely switch to the evil side. And I would say that this could be a very interesting concept.

However, the reader would probably feel cheated and start the eye-rolling if he was an innocent Luke-Skywalking paladin who never before hurt a fly, all of a sudden saying,
"Actually, yeah, it would be fun to rule the world with a genocidal iron fist alongside you! I bagsie Australia!".

Instead, there should be sort of clues leading to what's going to happen. Maybe show that he's prone to moments of greed or hypocrisy. Or misfortunes or tragedies happen that increase his level of cynicism, or he begins to notice the benefits of being greedy or treacherous.

It could be that his earlier idealistic goodness is borne out of a sense of naivety, but that same naivety causes him to turn evil when he fails consider the facts / all the balance.

Then, when he gets to the villain and the villain gives the big "You should be evil like me" speech that's what he needs to push him over the edge and a become full-blown evil henchman. Maybe he was feeling guilty about his immoral thoughts but the villain validates them for him and alleviates the guilt.

No Problem. This is the protagonist arc of a TRAGEDY, in which the hero fails to achieve his goal. One overarching trope of tragedy is that each of carries the seeds of our own destruction.

What is you character's flaw? How does he go about trying to satisfy the internal need by pursuing external goals? What happens that he fails (we all fail, they all fail, that's what Act II is made of), and why does ho persist in chasing the wrong idea?

Do you want the reader to agree with your protagonist at the end? Or should the reader revile the treacherous scum? It may be that the protagonist decides that it is necessary to throw away his principles, his morals, his values, his friends, family and loves -- his very self -- in order to accomplish the ONE THING.

So what is the ONE THING? Does he turn to evil as a noble sacrifice (that others may not be tainted, "sin-eater"), as a substitution of priorities, or is he eaten away, consumed by bitterness, spiting himself, the universe, the very notion of good and evil in order to make a final mark, and indelible wound upon the cosmos?

Was he the guy who always did the right thing and burnt out? And who warned him of his fate along the way, but Dudley Do-Right was too assured of his own moral strength?

As others have pointed out, it can be done, but it must be done well. "Happy endings" are not required, but they are easy.

It happened to the Three Musketeers, and they weren't the worse for it!

At the first, they were sworn enemies of Cardinal Richelieu. But at the end, the Cardinal offers D'Artagnan a promotion which he does indeed accept by virtue of the various circumstances in the well-told story.

Was D'Artagnan a defector? That's for you the reader to judge. Does it wreck the story?

I believe Star Wars story of Anakin Skywalker and Darth Sidious answers your question. The former eventually betrayed the Jedi Order and became a Sith Lord Darth Vader, an apprentice of the latter. It wasn't disappointing at all and nobody felt cheated I believe.