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. If these terms are large enough, they can push the system from an initial stable equilibrium point, over a nearby saddle point, outside of the basin of attraction of the stable point. Except in some specific cases, the distance between these two points is not known analytically. Focusing on Kuramoto-like models and under simple assumptions on this distance, 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. We numerically confirm the validity of that criterion for coupled oscillators on four very different complex networks. 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.