We present the first gas dynamical simulations of the photoevaporation of cosmological minihalos overtaken by the ionization fronts which swept through the IGM during reionization in a \Lambda CDM universe , including the effects of radiative transfer . We demonstrate the phenomenon of I-front trapping inside minihalos , in which the weak , R-type fronts which traveled supersonically across the IGM decelerated when they encountered the dense , neutral gas inside minihalos , becoming D-type I-fronts , preceded by shock waves . For a minihalo with virial temperature T _ { vir } \leq 10 ^ { 4 } K , the I-front gradually burned its way through the minihalo which trapped it , removing all of its baryonic gas by causing a supersonic , evaporative wind to blow backwards into the IGM , away from the exposed layers of minihalo gas just behind the advancing I-front . Such hitherto neglected feedback effects were widespread during reionization . N-body simulations and analytical estimates of halo formation suggest that sub-kpc minihalos such as these , with T _ { vir } \leq 10 ^ { 4 } K , were so common as to dominate the absorption of ionizing photons . This means that previous estimates of the number of ionizing photons per H atom required to complete reionization which neglected this effect may be too low . Regardless of their effect on the progress of reionization , however , the minihalos were so abundant that random lines of sight thru the high- z universe should encounter many of them , which suggests that it may be possible to observe the processes described here in the absorption spectra of distant sources .