To obtain direct measurements of the muon content of extensive air showers with energy above $10^{16.5}$ eV, the Pierre Auger Observatory is currently being equipped with an underground muon detector (UMD), consisting of 219 10 $\mathrm{m^2}$-modules, each segmented into 64 scintillators coupled to silicon photomultipliers (SiPMs). Direct access to the shower muon content allows for the study of both of the composition of primary cosmic rays and of high-energy hadronic interactions in the forward direction. As the muon density can vary between tens of muons per m$^2$ close to the intersection of the shower axis with the ground to much less than one per m$^2$ when far away, the necessary broad dynamic range is achieved by the simultaneous implementation of two acquisition modes in the read-out electronics: the binary mode, tuned to count single muons, and the ADC mode, suited to measure a high number of them. In this work, we present the end-to-end calibration of the muon detector modules: first, the SiPMs are calibrated by means of the binary channel, and then, the ADC channel is calibrated using atmospheric muons, detected in parallel to the shower data acquisition. The laboratory and field measurements performed to develop the implementation of the full calibration chain of both binary and ADC channels are presented and discussed. The calibration procedure is reliable to work with the high amount of channels in the UMD, which will be operated continuously, in changing environmental conditions, for several years.