The technological appeal of multiferroics is the ability to control magnetism with electric field. For devices to be useful, such control must be achieved at room temperature. The only single-phase multiferroic material exhibiting unambiguous magnetoelectric coupling at room temperature is BiFeO3 (refs 4 and 5). Its weak ferromagnetism arises from the canting of the antiferromagnetically aligned spins by the Dzyaloshinskii-Moriya (DM) interaction. Prior theory considered the symmetry of the thermodynamic ground state and concluded that direct 180-degree switching of the DM vector by the ferroelectric polarization was forbidden. Instead, we examined the kinetics of the switching process, something not considered previously in theoretical work. Here we show a deterministic reversal of the DM vector and canted moment using an electric field at room temperature. First-principles calculations reveal that the switching kinetics favours a two-step switching process. In each step the DM vector and polarization are coupled and 180-degree deterministic switching of magnetization hence becomes possible, in agreement with experimental observation. We exploit this switching to demonstrate energy-efficient control of a spin-valve device at room temperature. The energy per unit area required is approximately an order of magnitude less than that needed for spin-transfer torque switching. Given that the DM interaction is fundamental to single-phase multiferroics and magnetoelectrics, our results suggest ways to engineer magnetoelectric switching and tailor technologically pertinent functionality for nanometre-scale, low-energy-consumption, non-volatile magnetoelectronics.