Some people may disagree that this is an example per se, but I'd put up the Feynman path integral (see Wikipedia), because:
it provided a completely new physical picture of quantum mechanics
it led to systematic development of quantum field theory and string theories, both of which have had led to enormous synergistic growth in mathematics
it uncovered a fundamental similarity between of stochastic processes and deterministic quantum dynamics
it used the connection between Lie algebras and Lie groups in new and unexpected ways
questions about the path measure have stimulated much development in measure theory and analysis
tricks like the Wick rotation not only relate statistical mechanics and quantum mechanics (and the corresponding field theories) to each other, but also have stimulated further research in applications of analytic continuation
...and probably more that I am unaware of.