In a similar vein, I also try to see it as an opportunity to make my conversation partner smarter (if I happen to be right, of course).

For "winning", it's enough to prove that what they're saying is wrong. But for making them smarter, you need to point out what's correct and why that makes sense.
Well, and in general, it's a whole different way of formulating, i.e. less hostile, more helpful.

In the vast majority of cases, that makes all the difference for actually convincing them.
And it certainly hones your own mind much better, too, when you actually give the explanation rather than just pointing out fallacies.