Many young extra-galactic clusters have a measured velocity dispersion that is too high for the mass derived from their age and total luminosity, which has led to the suggestion that they are not in virial equilibrium. Most of these clusters are confined to a narrow age range centred around 10 Myr because of observational constraints. At this age the cluster light is dominated by luminous evolved stars, such as red supergiants, with initial masses of ~13-22 Msun for which (primordial) binarity is high. In this study we investigate to what extent the observed excess velocity dispersion is the result of the orbital motions of binaries. We demonstrate that estimates for the dynamical mass of young star clusters, derived from the observed velocity dispersion, exceed the photometric mass by up-to a factor of 10 and are consistent with a constant offset in the square of the velocity dispersion. This can be reproduced by models of virialised star clusters hosting a massive star population of which ~25 is in binaries, with typical mass ratios of ~0.6 and periods of ~1000 days. We conclude that binaries play a pivotal role in deriving the dynamical masses of young (~10 Myr) moderately massive and compact (≤ 5 Msun; > 1 pc) star clusters.