Many dwarf galaxies exhibit subsolar metallicities, with some star-to-star variation, despite often containing multiple generations of stars. The total metal content in these systems is much less than expected from the heavy-element production of massive stars in each episode of star formation. Such a deficiency implies that a substantial fraction of the enriched material has been lost from these small galaxies. Mass ejection from dwarf galaxies may have important consequences for the evolution of the intergalactic medium and for the evolution of massive galaxies, which themselves may have formed via the merger of smaller systems. We report here the results of three-dimensional simulations of the evolution of supernova-enriched gas within dwarf spheroidal galaxies (dSphs), with the aim of determining the retention efficiency of supernova ejecta. We consider two galaxy models, selected to represent opposite ends of the dSph sequence. One contains 10(6) M. of gas, and the other 5.5 x 10(6) M.. In both, the baryonic-to-dark matter ratio is assumed to be 0.1. The total binding energies of the gas in the two models are 9.8 x 10(50) and 1.6 x 10(52) ergs. For each model galaxy we investigate a number of scenarios. The simplest is a single supernova within a smooth gas distribution. We also investigate the evolution of 10 supernovae, within initially smooth gas distributions, occurring over time spans of either 10 or 100 Myr. Finally, we investigate the effects of 10 supernovae occurring over 10 Myr in a medium filled with hot "bubbles,'' such as would be expected in the presence of an initial generation of hot stars. For models with only a single supernova, no enriched material is lost from the galaxies. When multiple supernovae occur within an initially smooth gas distribution, less than one-half the enriched gas is lost from the galaxy (fractional losses range from 0% to 47%). Most of the enriched gas is lost, however, from the cores of the galaxies. In the presence of an initially disturbed gas distribution, 6% or less of the enriched gas remains in the core, and much is lost from the galaxies as a whole (47% and 71% for the larger and smaller galaxy models, respectively). If subsequent star formation occurs predominantly within the core where most of the residual gas is concentrated, then these results could explain the poor self-enrichment efficiency observed in dwarf galaxies.