Complex physical systems are unavoidably subjected to external environments not accounted for in the set of differential equations that models them. The resulting perturbations are standardly represented by noise terms. We derive conditions under which such noise terms perturb the dynamics strongly enough that they lead to stochastic escape from the initial basin of attraction of an initial stable equilibrium state of the unperturbed system. Focusing on Kuramoto-like models we find in particular that, quite counterintuitively, systems with inertia leave their initial basin faster than or at the same time as systems without inertia, except for strong white-noise perturbations.