Cantor's proof of the existence of transcendental numbers. With a (now) obvious one-line argument he showed that there are uncountably many of them --- when Liouville, Hermite and others had to take (putative) transcendental numbers one at a time ...
Newman's argument (especially Korevaar's and Zagier's version of it) turned the Prime Number Theorem, which took a century to be proved, into something that can be explained in a few minutes to any graduate student.