What newer mathematics fields helped to solve or solved problems from older fields of mathematics?
Solution 1:
Fermat's Last Theorem exemplifying how the theory of elliptic curves solves a classical problem in number theory.
The proof of the fundamental theorem of algebra using homotopy theory, showing how homotopy theory solves a problem in analysis.
Robinson's formalism of nonstandard analysis, showing how logic settles a long standing debate about the existence of infinitesimals.
Kolmogorov's axiomatization of probability theory, showing how measure theory is used to give a sound foundation for probability theory.
Numerous results in group theory that are proved using Shelah's classification theory, showing how logic solves problems in algebra.
Shannon's notion of entropy that created the whole field of information theory, shattering common beliefs in coding theory (e.g., Shannon's theorem on channel capacity).
I'm sure the list continues....
Solution 2:
Probably the most famous (recent) example is the theory of elliptic curves which was used by Andrew Wiles to prove Fermat's Last Theorem, a conjecture about a particular class of diophantine equations. The problem was famously scrawled in the margin of Fermat's copy of Diophantus' Arithmetica with a claim that Fermat had found a proof. However, Fermat's claimed proof was never found and it took over $350$ years to find one using machinery that had not been invented at the time.
There is a lot to this story; I recommend the BBC documentary if you would like to know more
Solution 3:
Galois theory
proves the impossibility of classical construction problems with compass and straightedge. (squaring the circle, doubling the cube, angle trisection). Moreover it gives an exact criterion for the constructibility of regular polygons.
characterizes the solvability of polynomial equations with radicals.
Solution 4:
Distribution theory, invented about 1950, is now extremely important in the modern theory of linear partial differential equations, e.g. the Malgrange-Ehrenpreis theorem or regularity theorems for elliptic PDO's