I related myself to this situation and pondered over it and came to the conclusion that even I sometimes do the same thing. If you also give a little thought to this, then you may also realize the same.
The question then arises is, Is it really necessary to win every argument?
Battle or War?
If the argument is heading in the direction where none amongst the two is really willing to understand the situation, then two scenarios may arise.
One, you are totally sure that your point is valid but still the argument is going in an absurd direction. If the other person is arguing insensibly and you've realized that your efforts to make your point out of logic will be wasted, then the wiser way would be to leave the argument at that time. (Keep in mind that you are later going to show the other person that he was wrong). Then later at the right time, when the other person is in the state of mind of understanding your words, then talk to him and make him realize his mistake. You may also use your actions to demonstrate how the other person was wrong (Actions speak louder than words!). Remember, and argument is not about concluding who's right and who's wrong, but about coming to a common understanding ground.
The other scenario can be, that in the middle of the argument you realize that you are wrong. I know this is an awful moment, but remember, it is not about if you're right or wrong. It is about making the other person understand you and also at the same time extending yourself to understand the other person. If you accept that you were wrong, when you really were, then this will not make you a loser, but it will show that you are a mature and wise person and in turn will earn more respect for you. Moreover, winning a heart is more important than winning an argument, especially if the argument is with your loved ones.
So next time you are in an argument, just ask yourself, "Am I arguing to come to a common solution or just to make a point?"