The activation problem is investigated in two-dimensional nonequilibrium systems. A numerical approach based on dynamic importance sampling (DIMS) is introduced. DIMS accelerates the simulations and allows the investigation to access noise intensities that were previously forbidden. The escape path is observed to be shifted compared to a heteroclinic trajectory calculated in the limit of zero-noise intensity. A theory to account for such shifts is presented and shown to agree with the simulations for a wide range of noise intensities.