The use of electric fields applied across magnetic heterojunctions that lack spatial inversion symmetry has been previously proposed as a nonmagnetic means of controlling localized magnetic moments through spin-orbit torques (SOT). The implementation of this concept at the single-molecule level has remained a challenge, however. Here, we present first-principles calculations of SOT in a single-molecule junction under bias and beyond linear response. Employing a self-consistency scheme invoking density functional theory and nonequilibrium Green's function theory including spin-orbit interaction, we compute the change of the magnetization with the bias voltage and the associated current-induced SOT. Within the linear regime our quantitative estimates for the SOT in single-molecule junctions yield values similar to those known for magnetic interfaces. Our findings contribute to an improved microscopic understanding of SOT in single molecules.