We present a technique to extract radial velocity measurements from echelle spectrograph observations of rapidly rotating stars ($V\sin{i} \gtrsim 50$ km s$^{-1}$). This type of measurement is difficult because the line widths of such stars are often comparable to the width of a single echelle order. To compensate for the scarcity of lines and Doppler information content, we have developed a process that forward-models the observations, fitting the radial velocity shift of the star for all echelle orders simultaneously with the echelle blaze function. We use our technique to extract radial velocity measurements from a sample of rapidly rotating A- and B-type stars used as calibrator stars observed by the California Planet Survey observations. We measure absolute radial velocities with a precision ranging from 0.5-2.0 km s$^{-1}$ per epoch for more than 100 A- and B-type stars. In our sample of 10 well-sampled stars with radial velocity scatter in excess of their measurement uncertainties, three of these are single-lined binaries with long observational baselines. From this subsample, we present detections of two previously unknown spectroscopic binaries and one known astrometric system. Our technique will be useful in measuring or placing upper limits on the masses of sub-stellar companions discovered by wide-field transit surveys, and conducting future spectroscopic binarity surveys and Galactic space-motion studies of massive and/or young, rapidly-rotating stars. ; Comment: Accepted to ApJS