A wonderful example is the proof of the Poincare Lemma I sketch here, as compared to the proof in e.g. Spivak's Calculus on Manifolds. The latter is extremely computational and, IIRC, not illuminating; it proves the de Rham cohomology of a star-shaped domain vanishes. The former proof shows (the stronger result) that the de Rham complex $\Lambda_{DR}(M)$ is null-homotopic for $M$ contractible; while this does involve some computation, it is very simple and conceptual. The proof is about half a page long in total, and could probably be shortened. It was shown to me by Professor Dennis Gaitsgory; I haven't seen it elsewhere, though I'm sure it is in the literature.

You can skip to the end of the paper (page 26) to see the proof; much of it is aimed at an undergraduate audience that has not yet seen any homological algebra, or even more conceptual linear algebra.

Essentially, the proof works by 1) Noting that the de Rham complex construction is functorial, via pullback of differential forms; 2) Noting that a homotopy of maps $M\to N$ induces a homotopy of maps of chain complexes; and 3) Noting that for $M$ contractible, $\operatorname{id}_M$ is homotopic to a constant map, and thus the pullback via $\operatorname{id}_M$ is both zero and the identity on cohomology.