The spin Hall effect in heavy metals converts charge current into pure spin current, which can be injected into an adjacent ferromagnet to exert a torque. This spin-orbit torque (SOT) has been widely used to manipulate the magnetization in metallic ferromagnets. In the case of magnetic insulators (MIs), although charge currents cannot flow, spin currents can propagate, but current-induced control of the magnetization in a MI has so far remained elusive. Here we demonstrate spin-current-induced switching of a perpendicularly magnetized thulium iron garnet film driven by charge current in a Pt overlayer. We estimate a relatively large spin-mixing conductance and damping-like SOT through spin Hall magnetoresistance and harmonic Hall measurements, respectively, indicating considerable spin transparency at the Pt/MI interface. We show that spin currents injected across this interface lead to deterministic magnetization reversal at low current densities, paving the road towards ultralow-dissipation spintronic devices based on MIs.