We present the first limits on inelastic electron-scattering dark matter and dark photon absorption using a prototype SuperCDMS detector having a charge resolution of 0.1 electron-hole pairs (CDMS HVeV, a 0.93 gram CDMS HV device). These electron-recoil limits significantly improve experimental constraints on dark matter particles with masses as low as 1 MeV/$\mathrm{c^2}$. We demonstrate a sensitivity to dark photons competitive with other leading approaches but using substantially less exposure (0.49 gram days). These results demonstrate the scientific potential of phonon-mediated semiconductor detectors that are sensitive to single electronic excitations.