Although mantle slabs ultimately drive plate motions, the mechanism by which they do so remains unclear. A detached slab descending through the mantle will excite mantle flow that exerts shear tractions on the base of the surface plates. This slab suction force drives subducting and overriding plates symmetrically toward subduction zones. Alternatively, cold, strong slabs may effectively transmit stresses to subducting surface plates, exerting a direct slab pull force on these plates, drawing them rapidly toward subduction zones. This motion induces mantle flow that pushes overriding plates away from subduction zones. We constrain the relative importance of slab suction and slab pull by comparing Cenozoic plate motions to model predictions that include viscous mantle flow and a proxy for slab strength. We find that slab pull from upper mantle slabs combined with slab suction from lower mantle slabs explains the observation that subducting plates currently move ~4 times faster than nonsubducting plates. This implies that upper mantle slabs are strong enough to support their own weight. Slab suction and slab pull presently account for about 40 and 60% of the forces on plates, but slab suction only ~30% if a low-viscosity asthenosphere decouples plates from mantle flow. The importance slab pull has been increasing steadily through the Cenozoic because the mass and length of upper mantle slabs has been increasing. This causes subducting plates to double their speed relative to nonsubducting plates during this time period. Our model explains this temporal evolution of plate motions for the first time.