We show that the often unsatisfactory performance of Møller-Plesset second-order perturbation theory (MP2) for the dispersion interaction between closed-shell molecules can be rectified by adding a correction Delta C(n)/R(n), to its long-range behavior. The dispersion-corrected MP2 (MP2 + Delta vdW) results are in excellent agreement with the quantum chemistry "gold standard" [coupled cluster theory with single, double and perturbative triple excitations, CCSD(T)] for a range of systems bounded by hydrogen bonding, electrostatics and dispersion forces. The MP2 + Delta vdW method is only mildly dependent on the short-range damping function and consistently outperforms state-of-the-art dispersion-corrected density-functional theory.