The purity of math is in the eye of the beholder, but maybe some of the following examples qualify:

There is a proof of Kolmogorov's strong law of large numbers in the book Probability and Finance by Shafer and Vovk that uses the determinacy of quasi-Borel games. The book generally shows how one can use game theory to prove probabilistic results.

Versions of the minimax theorem can be used to prove results in convex analysis. There is even a journal called Minimax Theory and its Applications. It is probably the most useful mathematical tool that came out of game theory.

Konrad Podczeck and I have a purification theorem for measure-valued maps whose proof is at least heavily based on game theoretic intuitions.

I would say the most useful applications of game theory to other areas of mathematics are based on zero-sum games, which are of least interest from the perspective of game theory as a tool of social sciences.