Milagro observations have found bright, diffuse TeV emission concentrated along the galactic plane of the Milky Way. The intensity and spectrum of this emission is difficult to explain with current models of hadronic γ-ray production, and has been named the "TeV excess." We show that TeV emission from pulsars naturally explains this excess. Recent observations have detected "TeV halos" surrounding pulsars that are either nearby or particularly luminous. Extrapolating this emission to the full population of Milky Way pulsars indicates that the ensemble of "subthreshold" sources necessarily produces bright TeV emission diffusively along the Milky Way plane. Models indicate that the TeV halo γ-ray flux exceeds that from hadronic γ rays above an energy of ∼500 GeV. Moreover, the spectrum and intensity of TeV halo emission naturally matches the TeV excess. Finally, we show that upcoming HAWC observations will resolve a significant fraction of the TeV excess into individual TeV halos, conclusively confirming, or ruling out, this model.