Some notable examples that come to mind are calculus, conservation laws, Dirac's delta and bra-ket notation in quantum mechanics, path integrals, renormalization.
Calculus is not a fair example, because the disciplines of mathematics and physics were not separated at the time; the work of Newton and Leibniz was the best "mathematics" (as well as "physics") of the era. Noether's theorem is not an example at all but a counterexample: a mathematical theorem, proved by a mathematician, that provided new insights into physics.
Dirac's delta and Feynman path integrals are fair examples of your point.
I concede the calculus point on a technicality :). Certainly there was no clear math/physics separation at the time (and the term physics as currently understood didn't even exist), but the drive to develop the math necessary to solve real-life problems was certainly there, and separate from the drive to do pure mathematics. And it took a long time before the d/dx notation was properly formalized.
As for the Noether's theorem, it was inspired by Einstein proving the energy-momentum tensor conservation in General Relativity, without realizing that it was a special case of a very general principle.
In Less Wrong Rationality and Mainstream Philosophy, Conceptual Analysis and Moral Theory, and Pluralistic Moral Reductionism, I suggested that traditional philosophical conceptual analysis often fails to be valuable. Neuroscientist V.S. Ramachandran has recently made some of the points in a polite sparring with philosopher Colin McGinn over Ramachandran's new book The Tell-Tale Brain: