Abstract Fast radio bursts (FRBs) are bright radio transients of microsecond to millisecond duration and unknown extragalactic origin. Central to the mystery of FRBs are their extremely high characteristic energies, which surpass the typical energies of other radio transients of similar duration, like Galactic pulsar and magnetar bursts, by orders of magnitude. Calibration of FRB-detecting telescopes for burst flux and fluence determination is crucial for FRB science, as these measurements enable studies of the FRB energy and brightness distribution in comparison to progenitor theories. The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a radio interferometer of cylindrical design. This design leads to a high FRB detection rate but also leads to challenges for CHIME/FRB flux calibration. This paper presents a comprehensive review of these challenges, as well as the automated flux calibration software pipeline that was developed to calibrate bursts detected in the first CHIME/FRB catalog, consisting of 536 events detected between 2018 July 25 and 2019 July 1. We emphasize that, due to limitations in the localization of CHIME/FRB bursts, flux and fluence measurements produced by this pipeline are best interpreted as lower limits, with uncertainties on the limiting value.