The EXtreme-PREcision Spectrograph (EXPRES) is an environmentally stabilized, fiber-fed, R = 137,500, optical spectrograph. It was recently commissioned at the 4.3 m Lowell Discovery Telescope near Flagstaff, Arizona. The spectrograph was designed with a target radial-velocity (RV) precision of 30 cm s−1. In addition to instrumental innovations, the EXPRES pipeline, presented here, is the first on-sky, optical, fiber-fed spectrograph to employ many novel techniques—including an “extended flat” fiber used for wavelength-dependent quantum efficiency characterization of the CCD, a flat-relative optimal extraction algorithm, chromatic barycentric corrections, chromatic calibration offsets, and an ultra-precise laser frequency comb for wavelength calibration. We describe the reduction, calibration, and RV analysis pipeline used for EXPRES and present an example of our current sub-meter-per-second RV measurement precision, which reaches a formal, single-measurement error of 0.3 m s−1 for an observation with a per-pixel signal-to-noise ratio of 250. These velocities yield an orbital solution on the known exoplanet host 51 Peg that matches literature values with a residual rms of 0.895 m s−1.