The Astrophysical Journal Letters 763.2 (2013): L41 reproduced by permission of the AAS ; We use a cosmological simulation of the formation of the Local Group of Galaxies to identify a mechanism that enables the removal of baryons from low-mass halos without appealing to feedback or reionization. As the Local Group forms, matter bound to it develops a network of filaments and pancakes. This moving web of gas and dark matter drifts and sweeps a large volume, overtaking many halos in the process. The dark matter content of these halos is unaffected but their gas can be efficiently removed by ram pressure. The loss of gas is especially pronounced in low-mass halos due to their lower binding energy and has a dramatic effect on the star formation history of affected systems. This "cosmic web stripping" may help to explain the scarcity of dwarf galaxies compared with the numerous low-mass halos expected in ΛCDM and the large diversity of star formation histories and morphologies characteristic of faint galaxies. Although our results are based on a single high-resolution simulation, it is likely that the hydrodynamical interaction of dwarf galaxies with the cosmic web is a crucial ingredient so far missing from galaxy formation models.