Young loggerhead sea turtles (Caretta caretta) from eastern Florida, USA, undertake a transoceanic migration in which they gradually circle the Sargasso Sea before returning to the North American coast. Loggerheads possess a ‘magnetic map’ in which regional magnetic fields elicit changes in swimming direction along the migratory pathway. In some geographic areas, however, ocean currents move more rapidly than young turtles can swim. Thus, the degree to which turtles can control their migratory movements has remained unclear. In this study, the movements of young turtles were simulated within a high-resolution ocean circulation model using several different behavioral scenarios, including one in which turtles drifted passively and others in which turtles swam briefly in accordance with experimentally derived data on magnetic navigation. Results revealed that small amounts of oriented swimming in response to regional magnetic fields profoundly affected migratory routes and endpoints. Turtles that engaged in directed swimming for as little as 1–3 h per day were 43–187% more likely than passive drifters to reach the Azores, a productive foraging area frequented by Florida loggerheads. They were also more likely to remain within warm-water currents favorable for growth and survival, avoid areas on the perimeter of the migratory route where predation risk and thermal conditions pose threats, and successfully return to the open-sea migratory route if carried into coastal areas. These findings imply that even weakly swimming marine animals may be able to exert strong effects on their migratory trajectories and open-sea distributions through simple navigation responses and minimal swimming.