I would nominate the definition of probability theory in terms of measure theory, where a probability space is an abstract measure space, an event is a measurable subset, a random variable is a measurable function, and expectation is the Lebesgue integral.  This is usually credited to a 1933 paper by Kolmogorov, though it seems that many of the ideas had previously been around.  [Here](http://www.probabilityandfinance.com/articles/04.pdf) is a very interesting survey by Shafer and Vovk, entitled "The origins and legacy of Kolmogorov's *Grundbegriffe*."

People had been thinking about probability for a long time before that, but these definitions made it possible to place everything on a rigorous footing and exploit many key results from measure theory.  Shafer and Vovk say that it took probability from a mathematical "pastime" to become a respected branch of pure mathematics.

(I am no expert historian so please feel free to edit with corrections, more background, further discussion, etc.)