© 2016 American Physical Society. We compute the non-Gaussian contribution to the covariance of the matter power spectrum at one-loop order in standard perturbation theory (SPT), using the framework of the effective field theory (EFT) of large scale structure (LSS). The complete one-loop contributions are evaluated for the first time, including the leading EFT corrections that involve seven independent operators, of which four appear in the power spectrum and bispectrum. We compare the non-Gaussian part of the one-loop covariance computed with both SPT and EFT of LSS to two separate simulations. In one simulation, we find that the one-loop prediction from SPT reproduces the simulation well to ki+kj∼0.25 h/Mpc, while in the other simulation we find a substantial improvement of EFT of LSS (with one free parameter) over SPT, more than doubling the range of k where the theory accurately reproduces the simulation. The disagreement between these two simulations points to unaccounted for systematics, highlighting the need for improved numerical and analytic understanding of the covariance.