Widespread applications of magnetic devices require an efficient means to manipulate the local magnetization. One mechanism is the electrical spin-transfer torque associated with electron-mediated spin currents; however, this suffers from substantial energy dissipation caused by Joule heating. We experimentally demonstrated an alternative approach based on magnon currents and achieved magnon-torque-induced magnetization switching in Bi2Se3/antiferromagnetic insulator NiO/ferromagnet devices at room temperature. The magnon currents carry spin angular momentum efficiently without involving moving electrons through a 25-nanometer-thick NiO layer. The magnon torque is sufficient to control the magnetization, which is comparable with previously observed electrical spin torque ratios. This research, which is relevant to the energy-efficient control of spintronic devices, will invigorate magnon-based memory and logic devices.