Well, depending on the dilemma, if it's serious enough not to back down and I'm not talking about petty things I'm talking about life changing things. Then you stand your ground with respect, but if it's petty, humble yourself and let it go. Die to self. ;)
I get quiet and just know deep down inside that I am correct OR if it is something I can prove then I show him my proof.
If it's a point of fact that can be verified, I'm pretty quick to say ok, I don't remember it that way, but maybe. And then at the first opportunity I check the facts and tell him if I was right. I don't bother to tell him if I wasn't right. ;D If it's something that can't be verified, I act the same way but can't attempt to verify it. I've been wrong often enough - and right often enough - to realize the best tactic is to just be humble and then quickly race to Google and get an answer, when possible.
i say, "so , we agree to disagree" and change the subject. lol