We present numerical simulations of circumbinary accretion onto eccentric and circular binaries using the movingmesh code AREPO. This is the first set of simulations to tackle the problem of binary accretion using a finitevolume scheme on a freely moving mesh, which allows for accurate measurements of accretion onto individual stars for arbitrary binary eccentricity. While accretion onto a circular binary shows bursts with period of ∼5 times the binary period Pb, accretion onto an eccentric binary is predominantly modulated at the period ∼1Pb. For an equal-mass circular binary, the accretion rates onto individual stars are quite similar to each other, following the same variable pattern in time. By contrast, for eccentric binaries, one of the stars can accrete at a rate 10-20 times larger than its companion. This "symmetry breaking" between the stars, however, alternates over timescales of order 200Pb and can be attributed to a slowly precessing, eccentric circumbinary disk. Over longer timescales, the net accretion rates onto individual stars are the same, reaching a quasi-steady state with the circumbinary disk. These results have important implications for the accretion behavior of binary T Tauri stars and supermassive binary black holes.