Autonomous swimmers have been intensively studied in recent years due to their numerous potential applications in many areas ranging from biomedicine to environmental remediation. Their motion is based either on different self-propulsion mechanisms or on the use of various external stimuli. Herein, the synergy between the ion flux around self-electrophoretic Mg/Pt Janus swimmers and an external magnetic field is proposed as an efficient alternative mechanism to power swimmers on the basis of the resulting Lorentz force. A strong magnetohydrodynamic effect is observed due to the orthogonal combination of magnetic field and spontaneous ionic currents, leading to an increase of the swimmer speed by up to 2 orders of magnitude. Furthermore, the trajectory of the self-propelled swimmers can be controlled by the orientation of the magnetic field, due to the presence of an additional torque force caused by a horizontal cation flux along the swimmer edges, resulting in predictable clockwise or anticlockwise motion. In addition, this effect is independent of the swimmer size, since a similar type of rotational motion is observed for macro- and microscale objects.