Has everyone forgotten John Nash? I believe his contributions to game theory are among the most prominent examples of mathematical ideas which are widely used in other fields.
On the pure end, the De Giorgi-Nash(-Moser) theorem is a landmark of elliptic and parabolic PDE, which establishes the uniform control of solutions of linear PDE with no assumptions on the smoothness of the coefficients - this is widely used in applications to the nonlinear case. Nash also resolved the isometric embedding problem in differential geometry with a very clever analytic approach. The main part of his proof established a particular instance of what is now known as the Nash-Moser implicit function theorem. The statement is somewhat innocuous (following Richard Hamilton's formulation, it extends the implicit function theorem from Banach spaces to 'tame Frechet spaces') but Nash's proof was very daring. Nash also resolved the isometric embedding problem in a different way, by looking for a low-regularity solution. The 'impossible'-looking thing about this paper is that he shows that n(n+1)/2 simultaneous PDE can be satisfied by finding only n+2 different functions, the key being that this is impossible if the n+2 functions are even as much as twice-differentiable. Nash's proof is remarkably direct.
(I remember, when he got the Nobel Prize in 1994, sitting at a lunch table with a group of mathematicians in different fields, each of whom knew of Nash from his work in their subject, all trying to figure out whether "their" Nash was the guy getting the prize.)