We study the change in the magnetoexciton absorption induced by an electric field parallel to the plane of a shallow quantum well. We demonstrate that, for in-plane magnetic field orientation, the discrete confined exciton peak undergoes a transition into a continuum resonance. In contrast, for perpendicular magnetic fields, the exciton peak exhibits the usual Stark redshift. We show that such a transition originates from a resonant coupling between the discrete confined exciton state and the continuum of dark bulk exciton states. As a result, the confined exciton tunnels out of the quantum well as a whole without being ionized. We discuss the possible experimental applications of this effect. © 1999 The American Physical Society.