In a recent paper, Stoffer showed that, under a very weak restriction on the step size, weakly attractive invariant tori of dissipative perturbations of integrable Hamiltonian systems persist under symplectic numerical discretizations. Stoffer's proof works directly with the discrete scheme. Here, we show how such a result, together with approximation estimates, can be obtained by combining Hamiltonian perturbation theory and backward error analysis of numerical integrators. In addition, we extend Stoffer's result to dissipative perturbations of nonintegrable Hamiltonian systems in the neighborhood of a KAM toms.