I don't know how good an example this is. The Lefschetz fixed point theorem tells you that you can count (appropriately weighted) fixed points of a continuous function $f : X \to X$ from a compact triangulable space to itself by looking at the traces of the induced action of $f$ on cohomology. This is a powerful tool (for example it more-or-less has the Poincare-Hopf theorem as a special case).

Weil noticed that the number of points of a variety $V$ over $\mathbb{F}_{q^n}$ is the number of fixed points of the $n^{th}$ power of the Frobenius map $f$ acting on the points of $V$ over $\overline{\mathbb{F}_q}$ and, consequently, that it might be possible to describe the local zeta function of $V$ if one could write down the induced action of $f$ on some cohomology theory for varieties over finite fields. This led to the Weil conjectures, the discovery of $\ell$-adic cohomology, etc. I think this is a pretty good candidate for a powerful but unexpected technique.