We report spatially resolved measurements of diffuse baryons flowing into and out of the Milky Way (MW) galaxy, using archival Hubble Space Telescope spectra of bright background quasars. We spatially resolve these flows by investigating the gas properties in different regions of the sky. We co-add spectra of different background sources at close projected angular separation. This novel stacking technique dramatically increases the signal to noise of the spectra, allowing detection of low column density gas (log N > 12/cm2). We identify gas flows as inflowing/outflowing by using blue/redshifted high velocity absorption against the background spectra, respectively. We find that gas inflow/outflow rates of each region analyzed vary across the sky almost by an order of magnitude. On average, the MW is dominated by inflowing gas with a net gas inflow rate of ~0.2 Solar mass/year. This implies that the MW is accreting mass, which will fuel the formation of next generation of stars in the disk and provide crucial empirical constraints on Galactic evolutionary models.