The underground muon detector of the Pierre Auger Observatory is aimed at attaining direct measurements of the muonic component of extensive air showers produced by cosmic rays with energy from $10^{16.5}$ eV up to the region of the ankle (around $10^{18.7}$ eV). It consists of two nested triangular grids of underground scintillators with 433 m, and 750 m spacings and a total of 71 positions, each with 192 scintillator strips (30 m$^2$) deployed 2.3 m underground. The light produced by impinging muons in the scintillators is propagated with optical fibers towards an array of silicon photomultipliers. In this work, we present the development, validation, and performance of an end-to-end tool for simulating the response of the underground muon detector to single-muon signals, which constitutes the basis for further simulations of the whole array. Laboratory data and simulation outcomes are found consistent, showing that with the underground muon detector we can measure single muons, with an efficiency of 99 %, up to about 1050 particles arriving at exactly the same time in 30 m$^2$ of scintillator.
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