Empirical constraints on reionization require galactic ionizing photon escape fractions fesc>20%, but recent high-resolution radiation-hydrodynamic calculations have consistently found much lower values ~1-5%. While these models include strong stellar feedback and additional processes such as runaway stars, they almost exclusively consider stellar evolution models based on single (isolated) stars, despite the fact that most massive stars are in binaries. We re-visit these calculations, combining radiative transfer and high-resolution cosmological simulations from the Feedback in Realistic Environments (FIRE) project. For the first time, we use a stellar evolution model that includes a physically and observationally motivated treatment of binaries (the BPASS model). Binary mass transfer and mergers enhance the population of massive stars at late times (>3 Myr) after star formation, which in turn strongly enhances the late-time ionizing photon production (especially at low metallicities). These photons are produced after feedback from massive stars has carved escape channels in the ISM, and so efficiently leak out of galaxies. As a result, the time-averaged effective escape fraction (ratio of escaped ionizing photons to observed 1500 A photons) increases by factors 4-10, sufficient to explain reionization. While important uncertainties remain, we conclude that binary evolution may be critical for understanding the ionization of the Universe.