There are two famous examples from enumerative algebraic geometry. The Schubert calculus was used by Schubert to solve many elaborate enumerative problems, but it was only fairly recently that these results were verified according to modern standards of mathematical rigor. Also, string theory predicted some enumerative results that the mathematicians were only later able to verify.
More generally, modern theoretical physics has produced enormous numbers of mathematical results whose derivations are non-rigorous. Some of these have been rigorously verified but some remain open problems.
