The hypothesised main drivers of megafauna extinctions in the late Quaternary have wavered between over-exploitation by humans and environmental change, with recent investigations demonstrating more nuanced synergies between these drivers depending on taxon, spatial scale, and region. However, most studies still rely on comparing archaeologically based chronologies of timing of initial human arrival into naïve ecosystems and palaeontologically inferred dates of megafauna extinctions. Conclusions arising from comparing chronologies also depend on the reliability of dated evidence, dating uncertainties, and correcting for the low probability of preservation (Signor-Lipps effect). While some models have been developed to test the susceptibility of megafauna to theoretical offtake rates, none has explicitly linked human energetic needs, prey choice, and hunting efficiency to examine the plausibility of human-driven extinctions. Using the island of Cyprus in the terminal Pleistocene as an ideal test case because of its late human settlement (~ 14.2 ka–13.2 ka), small area (~ 11,000 km2), and low megafauna diversity (2 species), we developed stochastic models of megafauna population dynamics, with offtake dictated by human energetic requirements, prey choice, and hunting-efficiency functions to test whether the human population at the end of the Pleistocene could have caused the extinction of dwarf hippopotamus (Phanourios minor) and dwarf elephants (Palaeoloxodon cypriotes). Our models reveal not only that the estimated human population sizes (N = 3,000–7,000) in Late Pleistocene Cyprus could have easily driven both species to extinction within < 1,000 years, the model predictions match the observed, Signor-Lipps-corrected chronological sequence of megafauna extinctions inferred from the palaeontological record (Phanourios at ~ 12 ka–11.1 ka, followed by Palaeoloxodon at ~ 10.3 ka–9.1 ka).