The dwarf spheroidal satellite galaxies (dSphs) of the Milky Way are some of the most dark matter (DM) dominated objects known. We report on γ-ray observations of Milky Way dSphs based on six years of Fermi Large Area Telescope data processed with the new Pass8 event-level analysis. None of the dSphs are significantly detected in γ rays, and we present upper limits on the DM annihilation cross section from a combined analysis of 15 dSphs. These constraints are among the strongest and most robust to date and lie below the canonical thermal relic cross section for DM of mass ≲100 GeV annihilating via quark and τ-lepton channels.