Here are three keys to being more persuasive:
1.Make sure they know you respect their intelligence
2.Make sure they know you respect their character/intentions
3.Make sure they know that you're open to being wrong
These three things will instantly make you 1000x more persuasive with anybody.
*Edit: I realize in many cases you don't respect their intelligence or character, and you're 100% sure you're right, but that doesn't matter. If you want them to listen, you have to disarm their ego and you have to make them think you are open to being wrong. This accomplishes that. So bite the bullet and fake it. You won't be sorry.
Ask good questions. For those who haven't studied the classical model of education, use the five common topics of definition, comparison, relationship, circumstance, and testimony to continue the conversation. Logic will always win out, and by asking the questions, you are leading them to tear down their own argument.