We demonstrate through smoothed particle hydrodynamics simulations that a circumbinary disk can supply mass to the central binary through gas streams that penetrate the disk gap without closing it. The conditions for an efficient flow typically require the disk thickness-to-radius ratio z/r >~ 0.05, if the turbulent viscosity parameter alpha is greater than 0.01. This mass flow may be important for both the individual systems and their statistics. It occurs preferentially onto the lower mass object and acts toward equalization of component masses. The less massive component may be more luminous and easier to detect, owing to its larger accretion luminosity. For eccentric binaries, the mass flow is strongly modulated in time, providing diagnostics for both the disk and the binary. In the protostellar disks, the flow could be detected as shock emission phased with the binary orbit, resulting from stream impact with the circumstellar disks and/or the young stars. In the (super)massive black hole binaries in nuclei of galaxies, the flow may result from the surrounding interstellar medium and produce nearly periodic emission, as observed in quasar OJ 287. For star-planet-disk systems, our results show that the opening of a gap around a planet is not always sufficient for the termination of its growth. This suggests that planets supplied by gas streams from protoplanetary disks may outgrow Jupiter to become "superplanets" with properties heretofore reserved for stars.