The 1/N expansion of matrix models is asymptotic, and it requires non-perturbative corrections due to large N instantons. Explicit expressions for large N instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multi-cut case. We show that the recent exact results on topological string instanton amplitudes provide the non-perturbative contributions of large N instantons in generic multi-cut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its 1/N expansion, which we obtain at relatively high genus for a generic two-cut background. These results can be extended to certain non-conventional matrix models which admit a topological string theory description. As an application, we determine the large N instanton corrections for the free energy of ABJM theory on the three-sphere, which correspond to D-brane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold Gromov-Witten invariants. By focusing on the example of C³/Z₃, we show that they grow doubly-factorially with the genus and we obtain and test explicit asymptotic formulae for them.