We study the efficiency for producing strong gravitational lensing events of galaxy clusters numerically simulated in different dark-energy cosmologies with constant and time-variable equation of state, and we compare the results with those obtained in "standard" $Λ$CDM and OCDM models. Our main results are (1) that the expected abundance of gravitational arcs with large length-to-width ratio depends on the equation of state of dark energy at the epoch of formation of the halo; and (2) that the high sensitivity of strong-lensing cross sections of galaxy clusters to dynamical processes like mergers, which was found in earlier studies, varies substantially between different cosmologies, being stronger for models in which halos are less concentrated. As expected, the largest differences in the lensing optical depth occur at intermediate and high redshift.