Being Right Isn't Enough ...
... to convince someone of the truth.
I remember the look on one theologian's face when i asked about countering this tendency after he had given a talk on the rise of Neo-Modernism in the Church.
He simply thought that rationality would win the day.
If he had looked at political elections, he would have known that isn't really the case.
Rational thought only works when people are thinking rationally.
When they fall into irrationality, rational arguments only cause them to seize any irrational thought to defend their belief (see the cognitive dissonance model above).
So ... what to do?
First we have to help them maintain a rational state of mind.
Asking questions is probably the best way.
Have your interlocutor (definition) do the heavy lifting of thinking through their argument in a rational manner.
This way you don't directly challenge their misbegotten beliefs, causing a defensive retreat into irrationality.
The only drawback is this requires a great deal of patience.