Isn't almost every theorem in mathematics an example of a theorem "for free"? One defines natural numbers, and then it follows each of them is a sum of four squares; one defines a notion of a continuous function and of Euclidean space, and Brouwer's fixed point theorem follows. Surely, that is amazing!
With that said here are a handful of the example that lie closer to the surface:
1) Complex-differentiable functions are infinitely-differentiable, and in fact analytic.
2) A function of several complex variables that is holomorphic in each variable is holomorphic in all of them (if it reminds you of 'theorem' that a function that is continuous in each variable separately is continuous... well, then, it should). That is Hartogs' theorem.
3) Any bound on the error term in primes number theorem of the form $\psi(x)=x+O_{\varepsilon}(x^{a+\varepsilon})$ implies the bound $\psi(x)=x+O(x^a \log x)$.
4) Morally related to (3) is the tensor power trick, of which the earliest widely-known example is perhaps the proof of Cotlar-Stein lemma. One of my favorite examples is lemma 2.1 from a paper of Katz and Tao on Kakeya's conjecture.
