I suppose that this is an example that doesn't pack the historical weight of what many others have said, but topology has played an interesting (and up-and-coming) role in computer graphics and network sensing. This site details some of these applications and how they are currently changing various technologies. For example, there are a few examples of persistent homology being using to detect network coverage (i.e. Cell Tower Coverage) a la Ghrist and de Silva (2007) and for feature detection in MRI. Similarly, the ALICE project has used various Laplace-Beltrami operators for Computer Graphics applications as well as the use of Cell Complexes in surface reconstruction. On the other hand, there are examples of homology being used to characterize inviscid flows and in Computational Statistics / Statistical Mechanics (Chapter 5).
Again, these probably don't hold the weight of some of the other answers and at the moment, we don't know how these mathematical innovations will push technology forward; however, they represent the potential future for mathematics as a driving force for technology.
