Future information technologies, such as ultrafast data recording, quantum computation or spintronics, call for ever faster spin control by light1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. Intense terahertz pulses can couple to spins on the intrinsic energy scale of magnetic excitations5, 11. Here, we explore a novel electric dipole-mediated mechanism of nonlinear terahertz-spin coupling that is much stronger than linear Zeeman coupling to the terahertz magnetic field5, 10. Using the prototypical antiferromagnet thulium orthoferrite (TmFeO3), we demonstrate that resonant terahertz pumping of electronic orbital transitions modifies the magnetic anisotropy for ordered Fe3+ spins and triggers large-amplitude coherent spin oscillations. This mechanism is inherently nonlinear, it can be tailored by spectral shaping of the terahertz waveforms and its efficiency outperforms the Zeeman torque by an order of magnitude. Because orbital states govern the magnetic anisotropy in all transition-metal oxides, the demonstrated control scheme is expected to be applicable to many magnetic materials.