Galactic winds are a prime suspect for the metal enrichment of the intergalactic medium and may have a strong influence on the chemical evolution of galaxies and the nature of QSO absorption line systems. We use a sample of 1406 galaxy spectra at z~1.4 from the DEEP2 redshift survey to show that blueshifted Mg II 2796, 2803 A absorption is ubiquitous in starforming galaxies at this epoch. This is the first detection of frequent outflowing galactic winds at z~1. The presence and depth of absorption are independent of AGN spectral signatures or galaxy morphology; major mergers are not a prerequisite for driving a galactic wind from massive galaxies. Outflows are found in coadded spectra of galaxies spanning a range of 30x in stellar mass and 10x in star formation rate (SFR), calibrated from K-band and from MIPS IR fluxes. The outflows have column densities of order N_H ~ 10^20 cm^-2 and characteristic velocities of ~ 300-500 km/sec, with absorption seen out to 1000 km/sec in the most massive, highest SFR galaxies. The velocities suggest that the outflowing gas can escape into the IGM and that massive galaxies can produce cosmologically and chemically significant outflows. Both the Mg II equivalent width and the outflow velocity are larger for galaxies of higher stellar mass and SFR, with V_wind ~ SFR^0.3, similar to the scaling in low redshift IR-luminous galaxies. The high frequency of outflows in the star-forming galaxy population at z~1 indicates that galactic winds occur in the progenitors of massive spirals as well as those of ellipticals. The increase of outflow velocity with mass and SFR constrains theoretical models of galaxy evolution that include feedback from galactic winds, and may favor momentum-driven models for the wind physics.