Three-body and higher intermolecular interactions can play an important role in molecular condensed phases. Recent benchmark calculations found problematic behavior for many widely used density functional approximations in treating 3-body intermolecular interactions. Here, we demonstrate that the combination of second-order Møller-Plesset (MP2) perturbation theory plus short-range damped Axilrod-Teller-Muto (ATM) dispersion accurately describes 3-body interactions with reasonable computational cost. The empirical damping function used in the ATM dispersion term compensates both for the absence of higher-order dispersion contributions beyond the triple-dipole ATM term and non-additive short-range exchange terms which arise in third-order perturbation theory and beyond. Empirical damping enables this simple model to out-perform a non-expanded coupled Kohn-Sham dispersion correction for 3-body intermolecular dispersion. The MP2 plus ATM dispersion model approaches the accuracy of O(N(6)) methods like MP2.5 or even spin-component-scaled coupled cluster models for 3-body intermolecular interactions with only O(N(5)) computational cost.