Current numerical studies suggest that the first protogalaxies formed a few stars at a time and were enriched only gradually by the first heavy elements. However, these models do not resolve primordial supernova (SN) explosions or the mixing of their heavy elements with ambient gas, which could result in intervening, prompt generations of low-mass stars. We present multiscale 1D models of Population III supernovae in cosmological minihalos that evolve the blast from its earliest stages as a free expansion. We find that if the star ionizes the halo, the ejecta strongly interacts with the dense shell swept up by the H II region, potentially cooling and fragmenting it into clumps that are gravitationally unstable to collapse. If the star fails to ionize the halo, the explosion propagates metals out to 20 - 40 pc and then collapses, heavily enriching tens of thousands of solar masses of primordial gas, in contrast to previous models that suggest that such explosions ‘fizzle’. Rapid formation of low-mass stars trapped in the gravitational potential well of the halo is unavoidable in these circumstances. Consequently, it is possible that far more stars were swept up into the first galaxies, at earlier times and with distinct chemical signatures, than in present models. Upcoming measurements by the James Webb Space Telescope (JWST) and Atacama Large Millimeter Array (ALMA) may discriminate between these two paradigms.