We describe a measurement of the time-integrated luminosity of the data collected by the BABAR experiment at the PEP-II asymmetric-energy e+e- collider at the Upsilon(4S), Upsilon(3S), and Upsilon(2S) resonances and in a continuum region below each resonance. We measure the time-integrated luminosity by counting e+e- --> e+e- and (for the Upsilon(4S) only) e+e- --> mu+mu- candidate events, allowing additional photons in the final state. We use data-corrected simulation to determine the cross sections and reconstruction efficiencies for these processes, as well as the major backgrounds. Due to the large cross sections of e+e- --> e+e- and +e- --> mu+mu-, the statistical uncertainties of the measurement are substantially smaller than the systematic uncertainties. The dominant systematic uncertainties are due to observed differences between data and simulation, as well as uncertainties on the cross sections. For data collected on the Upsilon(3S) and Upsilon(2S) resonances, an additional uncertainty arises due to Upsilon --> e+e- background. For data collected off the Upsilon resonances, we estimate an additional uncertainty due to the time-dependent efficiency variations, which can affect the short off-resonance runs.