Unexpected use of topology in proofs
One of the richest sources for this is geometric group theory, and especially the parts of it that construct topological spaces corresponding to particular groups (schreier graphs, cayley complexes, BG spaces etc.) For example, the proof that any subgroup of a free group is free (Nielsen Schreier) epitomizes how topological arguments greatly simplify some parts of group theory.
There are proof methods for theorems from discrete geometry that utilize methods from equivariant topology (such as the ham-sandwich theorem via Borsuk ulam.) These use the CS/TM paradigm, which studies maps between the space of "geometric configurations" to the "Space of viable solutions."
See here for a general reference and wikipedia for a shorter exposition.
There are some other miscellaneous examples that I know of:
Bezout's identity via separating curves (assumes the euclidian algorithm in disguise.)
Fundamental Theorem of Algebra (resp. existence of eigenvalues) via lefschetz fixed point theorem
the use of "Denseness arguments" in algebraic geometry (such as cayley-hamilton).
How about the Nielsen-Schreier theorem? The statement of the theorem is entirely algebraic:
Every subgroup of a free group is free.
To prove it, we use algebraic topology: the free group on $n$ generators is the fundamental group of the bouquet of $n$ circles, and any subgroup of the fundamental group is the fundamental group of some covering space of this bouquet. Then it's a matter of using topological arguments to show that every covering space of the bouquet of $n$ circles is itself homotopy equivalent to some bouquet of circles.
One of the proofs of the Fundamental Theorem of Algebra uses topology.