We introduce an event-driven simulation scheme for overdamped dynamics of frictionless hard spheres subjected to external forces, neglecting hydrodynamic interactions. Our event-driven approach is based on an exact equation of motion which relates the driving force to the resulting velocities through the geometric information characterizing the underlying network of contacts between the hard spheres. Our method allows for a robust extraction of the instantaneous coordination of the particles as well as contact force statistics and dynamics, under any chosen driving force, in addition to shear flow and compression. It can also be used for generating high-precision jammed packings under shear, compression, or both. We present a number of additional applications of our method.