Some more proofs that startled me (in a random order):
Liouville theorem to prove that Weierstrass P-function satifies the differential equation you know.
Complex methods to establish the addition law on an elliptic curve.
Cauchy's formula (for P'/P) to prove that C is algebraically closed.
Pigeon hole principle to prove existence of solutions to Fermat-Pell's equation
Kronecker's solution to the same equation, using L-functions.
Minkowski's lemma (a convex compact, symmetric, of volume 2^n contains a non trivial integer point) and its use to prove Dirichlet's theorem on the structure of units in number fields.
Fourier transform to prove (versions of) the central limit theorem.
Multiplicativity of Ramanujan's tau function via Hecke operators.
Poisson formula and its use (for example, for the functional equation of Riemann's zeta function, or for computing the volume of SL_n(R)/SL_n(Z), or values of zeta at even positive integers).