We observed RX J0520.5-6932 in the X-rays and studied the optical light curve of its counterpart to verify it as a Be/X-ray binary. We performed an XMM-Newton anticipated target of opportunity observation in January 2013 during an X-ray outburst of the source in order to search for pulsations and derive its spectral properties. We monitored the source with Swift to follow the evolution of the outburst and to look for further outbursts to verify the regular pattern seen in the optical light curve with a period of ~24.4 d. The XMM-Newton EPIC light curves show coherent X-ray pulsations with a period of 8.035331(15) s (1 sigma). The X-ray spectrum can be modelled by an absorbed power law with photon index of ~0.8, an additional black-body component with temperature of ~0.25 keV and an Fe K line. Phase-resolved X-ray spectroscopy reveals that the spectrum varies with pulse phase. We confirm the identification of the optical counterpart within the error circle of XMM-Newton at an angular distance of ~0.8 arcsec, which is an O9Ve star with known Halpha emission. By analyzing the combined data from three OGLE phases we derived an optical period of 24.43 d.The X-ray pulsations and long-term variability, as well as the properties of the optical counterpart, confirm that RX J0520.5-6932 is a Be/X-ray binary pulsar in the Large Magellanic Cloud. Based on the X-ray monitoring of the source we conclude that the event in January 2013 was a moderately bright type-I X-ray outburst, with a peak luminosity of 1.79e36 erg/s.