We present a calculation of the matter power spectrum covariance matrix Cov (k 1, k 2) that uses power spectrum responses to accurately describe the coupling between large-and small-scale modes beyond the perturbative regime. These response functions can be measured with (small-volume) N-body simulations, which is why the response contributions to the covariance remain valid and predictive at all orders in perturbation theory. A novel and key step presented here is the use of responses to compute loop contributions with soft loop momenta, which extends the application of the response approach beyond that of previously considered squeezed n-point functions. The calculation presented here does not involve any fitting parameters. When including response-type terms up to 1-loop order in perturbation theory, we find that our calculation captures the bulk of the total covariance as estimated from simulations up to values of k 1, k 2∼ 1 h Mpc-1. Moreover, the prediction is guaranteed to be accurate whenever the softer mode is sufficiently linear,{k 1, k 1}≲ 0.08 h Mpc-1. We identify and discuss straightforward improvements in the context of the response approach, which are expected to further increase the accuracy of the calculation presented here.