Examples of "miraculous" proofs Concerning the proof that $\zeta(3)$ is irrational, Van der Poorten famously noted that
"Apéry's incredible proof appears to be a mixture of miracles and mysteries".
Indeed, many ideas introduced in Apéry's proof such as $\zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {\binom {2k}{k}k^{3}}}$ and the recurrence $n^3u_n + (n-1)^3 u_{n-2} = (34n^3-51n^2+27n-5)u_{n-1}, n\geq2$ amazed contemporary mathematicians (although the fast-converging series was already derived several years earlier by Hjortnaes).
What are other examples of "miraculous" proofs, whose ingredients amazed mathematicians of their time? In particular, proofs such that:


*

*a large portion of the Theorems involved in the proof represent entirely new ideas,

*these Theorems are applicable to a wide area of mathematics,

*the general atmosphere among contemporary mathematicians was a mixture of  surprise and awe ("where did this come from?").

 A: My favorite example is Gunther’s radically simpler proof of the Nash isometric embedding theorem, which came out of nowhere.
A: Gromov's theorem on groups with polynomial growth has, I think, an amazing proof. It's where he introduced Gromov-Hausdorff distance and showed, using the solution to Hilbert's fifth problem, that  a sequence of discrete groups, under the right assumptions, converges to a Lie group.
A: I am surprised that nobody has mentioned the Riemann hypothesis and its consequences. I can never hide my fascination that somebody could manage to prove that the distribution of prime numbers and many other problems in number theory, all boil down to proving that the roots of a seemingly nice function in the complex plane have equal real parts.
I wish I could some day travel back in time and ask him really, what kind of sorcery is this?
A: I think Stanley's introduction of tools from commutative algebra to study convex polytopes in his proof of the upper bound theorem for simplicial spheres was completely novel and miraculous; for a historical account see: http://www-math.mit.edu/~rstan/papers/ubc.pdf.
A: The best one I can think of is Paul Cohen's proof of the independence of the continuum hypothesis. That certainly had a huge impact, and I may be wrong, but my impression is that set theorists at the time felt that the proof came out of nowhere. In retrospect maybe one can find some hints of the technique in earlier work, but again, at the time I think it was seen as totally new.
A: Nash equilibrium. I don't think before that anyone guessed Analysis would have an application in Game Theory.
A: Going further back, Galois correspondence came out of nowhere.
A: Thurston's geometrization conjecture came out
of the blue around 1980. He proved it for Haken manifolds in 1982.
In his famous essay, On proof and progress in mathematics, he says:

"It was a hard theorem, and I spent
  a tremendous amount of effort thinking about it.
  ...
  Neither the geometrization conjecture nor its proof for Haken manifolds was in
  the path of any group of mathematicians at the time—it went against the trends
  in topology for the preceding 30 years, and it took people by surprise."

A: I nominate Jack Silver's proof that, if the generalized continuum hypothesis is false, then the first counterexample cannot be a singular cardinal of uncountable cofinality.  In the first place, the result was, as far as I know, totally unexpected. The general feeling was that Easton's results on violations of GCH at regular cardinals should extend to singular cardinals.  Secondly, the proof was an unexpected mixture of combinatorial set theory and nonstandard models. (Later, other proofs were found that avoid the nonstandard models.)
A: I think Donaldson's work on gauge-theoretic invariants for smooth 4-manifolds was also considered miraculous by topologists at first (as probably was Witten's introduction of SW invariants which are roughly equivalent in power to Donaldson invariants and MUCH easier to work with). 
A: I quote from Michael Monastyrsky's book Modern Mathematics in the Light of the Fields Medals.

The most important of Margulis' results is his proof of Selberg's conjecture that a certain class of discrete groups is arithmetic.  … Margulis' papers display his exceptional originality. By invoking ideas from widely separated branches of mathematics, he proceeds to the goal by the simplest possible route.  One of these theorems, which was central to proving the theorem on arithmetic subgroups, was announced several years before the full proof appeared.  Despite efforts of many leading experts in this area, the result could not be reproduced independently.  The nature of Margulis' original proof was very simple and, to the experts, unexpected.

A: It's not really a theorem and maybe not miraculous, but I think the way Gromov, in his book Partial Differential Relations solves an underdetermined system of PDEs (fewer equations than unknown functions) is hilarious. Normally, a differential equation is solved using integration in one way or another. Gromov shows that many sufficiently underdetermined systems can be solved explicitly in terms of  the coefficients and their derivatives. The simplest example is to solve for $u$ and $v$ satisfying
$$ \partial_x u + v = f $$
The simplest nontrivial example, just an ODE, is:
$$ au' + bv' = f $$
Note that if $a$ and $b$ are constant, then the only way to solve this is by integration. However, if $a$ and $b$ are functions satisfy a nondegeneracy condition, then a solution $(u,v)$ can be written explicitly in terms of $a$, $b$, $f$, and their derivatives. See my note on this.
A: I've always been impressed by Lovasz's cancellation theorem among finite relational structures. Some details are at https://mathoverflow.net/a/269610 and nearby.
Gerhard "Search MathOverflow For More Answers" Paseman, 2018.07.13.
