My favorite example from algebraic topology is Rene Thom's work on cobordism theory. The problem of classifying manifolds up to cobordism looks totally intractable at first glance. In low dimensions ($0,1,2$), it is easy, because manifolds of these dimensions are completely known. With hard manual labor, one can maybe treat dimensions 3 and 4. But in higher dimensions, there is no chance to proceed by geometric methods.

Thom came up with a geometric construction (generalizing earlier work by Pontrjagin), which is at the same time easy to understand and ingenious. Embed the manifold into a sphere, collapse everything outside a tubular neighborhood to a point and use the Gauss map of the normal bundle... What this construction does is to translate the geometric problem into a homotopy problem, which looks totally unrelated at first sight.

The homotopy problem is still difficult, but thanks to work by Serre, Cartan, Steenrod, Borel, Eilenberg and others, Thom had enough heavy guns at hand to get fairly complete results.

Thom's work led to an explosion of differential topology, leading to Hirzebruch's signature theorem, the Hirzebruch-Riemann-Roch theorem, Atiyah-Singer, Milnor-Kervaire classification of exotic spheres.....until Madsen-Weiss' work on mapping class groups.