Ultrafast charge and spin excitations in the elusive terahertz regime1, 2 of the electromagnetic spectrum play a pivotal role in condensed matter3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. The electric field of free-space terahertz pulses has provided a direct gateway to manipulating the motion of charges on the femtosecond timescale6, 7, 8, 9. Here, we complement this process by showing that the magnetic component of intense terahertz transients enables ultrafast control of the spin degree of freedom. Single-cycle terahertz pulses switch on and off coherent spin waves in antiferromagnetic NiO at frequencies as high as 1 THz. An optical probe pulse with a duration of 8 fs follows the terahertz-induced magnetic dynamics directly in the time domain and verifies that the terahertz field addresses spins selectively by means of the Zeeman interaction. This concept provides a universal ultrafast means to control previously inaccessible magnetic excitations in the electronic ground state.