Extensive air showers, originating from ultra-high energy cosmic rays, have been successfully measured through the use of arrays of water-Cherenkov detectors (WCDs). Sophisticated analyses exploiting WCD data have made it possible to demonstrate that shower simulations, based on different hadronic-interaction models, cannot reproduce the observed number of muons at the ground. The accurate knowledge of the WCD response to muons is paramount in establishing the exact level of this discrepancy. In this work, we report on a study of the response of a WCD of the Pierre Auger Observatory to atmospheric muons performed with a hodoscope made of resistive plate chambers (RPCs), enabling us to select and reconstruct nearly 600 thousand single muon trajectories with zenith angles ranging from 0$^\circ$ to 55$^\circ$. Comparison of distributions of key observables between the hodoscope data and the predictions of dedicated simulations allows us to demonstrate the accuracy of the latter at a level of 2%. As the WCD calibration is based on its response to atmospheric muons, the hodoscope data are also exploited to show the long-term stability of the procedure.