“He who wants to persuade should put his trust not in the right argument, but in the right word. The power of sound has always been greater than the power of sense.” - Joseph Conrad, Lord Jim, 1900.
Forget about arguments. They will come and go. But words last forever. Think fast but think twice about what you are going to say. When you start any conversation, even a debate, with a kind word you will find that you have gained the attention you need to put the right words.
If you had tried your best and the other party continues in arrogance - or ignorance, or both - think about this: "Never argue with a fool. Someone watching may not be able to tell the difference."
And when you leave a conversation (or a debate) let a kind word, a compliment, be your strong ending.