The proof of the Mumford conjecture by Madsen and Weiss made essential use of Teichm"uller theory. This becomes especially clear if you state the conjecture as being about the cohomology of the space $\mathfrak{M}_g$, Riemanns moduli space of genus $g$ complex curves. Defining this space does not require Teichmueller theory, it can be done in a purely algebro-geometric way.
The first step is that $H_{\ast} (\mathfrak{M}_g;\mathbb{Q})$ is the same as the rational homology of the mapping class group $\Gamma_g$. This uses Teichm"ullers theorem that Teichm"ullers space $\mathcal{T}_g$ is homeomorphic to a euclidean space (being a contractible manifold would be enough).
The second step is that $B \Gamma_g$ is homotopy equivalent to $B Diff (\Sigma_g)$, the classifying space of the diffeomorphism group. This is a result by Earle and Eells, which uses Teichm"ullers theorem as well, albeit not so essentially, because there is a purely topological proof of this result as well.
The Madsen-Weiss theorem then computes the homology of $B Diff (\Sigma_g)$ in a range of degrees; this is differential topology/homotopy theory and not related to Teichm"uller theory.
Older results on the homology of $\mathfrak{M}_g$ (or the Deligne-Mumford compactification) are very often based on step 1 as well. Some relevant names are Harer, Harer-Zagier, Arbarello-Cornalba- and others.