Electric-field control of magnetization promises to substantially enhance the energy efficiency of device applications ranging from data storage to solid-state cooling. However, the intrinsic linear magnetoelectric effect is typically small in bulk materials. In thin films, electric-field tuning of spin-orbit-interaction phenomena (e.g., magnetocrystalline anisotropy) has been reported to achieve a partial control of the magnetic state. Here we explore the piezomagnetic effect (PME), driven by frustrated exchange interactions, which can induce a net magnetization in an antiferromagnet and reverse its direction via elastic strain generated piezoelectrically. Our ab initio study of PME in Mn-based antiperovskite nitrides identified an extraordinarily large PME in Mn3SnN available at room temperature. We explain the magnitude of PME based on features of the electronic structure and show an inverse proportionality between the simulated zero-temperature PME and the magnetovolume effect at the magnetic (Néel) transition measured by Takenaka et al. in nine antiferromagnetic Mn3AN systems.