argue (one's) way out (of something)
To escape a problem or punishment by a clever or otherwise effective argument or explanation. How did she argue her way out of yet another speeding ticket? I am absolutely furious with you. There is no way you can just argue your way out this time.
argue one's way out of something
and argue one's way outto talk and get oneself free of a problem. You can't argue your way out of this! It's a problem, and there is no way that you can argue your way out.