I would say an example par excellence of this type of intuition can be found in the creation of Calculus. Newton and Leibniz were operating on a different level of thought, they had a sense for a certain type of approach to mathematics and physical phenomena, and they were able to implement this approach without the sometimes overwhelming fetters of rigor. They did not supply "modern" proofs for their assertions, but their ideas worked. The duo (although not collaboratively) developed an ideal theory of infinitesimals, a way to measure change as it approaches the "infinite beyond" and the instantaneous moment.
To be more concrete, take what some might term the cornerstone of calculus, the limit. To Newton, this idea sprung forth from intuition; it made sense as an a priori (not in the strictest philosophical sense) idea. One feels that one knows what a limit is, and it is enough work in the practical with specific functions (say, $ lim_{x\to3}2x^2=18)$. But to the modernist, this mathematical object had to be given some precise definition that accords with rigor; it is not enough to couch intuition in a phrase like "as x approaches". And so the $\epsilon-\delta$ definition of the limit was born (some attribute it the idea to Cauchy, but it was brought to its fullest form by Bolzano and Weierstrass).
The point is that Newton, with his fluxions, and Leibniz with his $dx$ and the like, tapped into a whole area of mathematics, a whole area of physical study, without the addition of rigor! But even more that that, their ideas continue to influence almost all of mathematical thought today. This is what comes to my mind when I think of intuition in math.
