If the partner in argument already knows he's wrong and doesn't want to admit it, remember that there is actually no reason he should be forced to admit it. You know it, he knows it, and... unless there is an audience to the argument, that accounts for everyone involved in the situation. Instead of being a slave to your ego and working to get him to admit he was wrong, you can become the voice of the subconscious mind. In doing so, one option is to offer him a way out:
[Someone unreasonably becomes angry and abusive]
Communication Ninja: “Wait a minute... maybe the reason you got so angry was because you care so much about Molly.”
(Maybe Molly has nothing to do with the situation, but if the partner in argument knows he was wrong he will quickly accept that idea and confirm it.)
Partner in Argument: "Yeah, I don't like anyone talking about her like that, and I know that's what set me off."
Communication ninja: “Well I'm glad I realized that before making a fool of myself. I was thinking about the whole situation the wrong way. I understand where you're coming from and I'm sorry we clashed like this.”
So the moral of the story is that getting the other guy to admit he is wrong is not real winning. That builds resentment that can't be helpful to your real purpose. You can win by suddenly “noticing” something that makes you “understand where he is coming from,” and you can admit that you had judged him prematurely and that now you “regret clashing with him this way.” And to take it to an excellent extreme you can even say, "I'm glad I realized this before making a fool of myself." To the conscious mind of the partner in argument, this sounds like you are taking all responsibility for the problem, but the subconscious mind hears something like this: "REALIZE, before making a fool of myself, thinking the wrong way, understand where you are coming from..."
If it's clear to you that the other guy already knows he was wrong, this powerful gesture of reconciliation can be a source of both healing and … winning. The other guy comes away from the argument with the knowledge that you are not someone who can be easily bullied, and he also feels that you suddenly understand him and genuinely like him. And if you are a good practitioner, you really do like him, just as you like life.