An underactuated two-dimensional translational oscillator with rotational actuator (2DTORA) consisting of an actuated rotational proof-mass and two unactuated translational carts is presented. Passivity-based control design is employed for 2DTORA based on its Euler–Lagrange structure and passivity property. Firstly, the dynamics of 2DTORA are derived based on Euler–Lagrange equations. Motivated by constructing a damped close-loop Euler–Lagrange system, the controller dynamics is designed to shape the potential energy and inject the required damping. As a result, the designed controller stabilizes the underactuated 2DTORA with the feedback of the rotational actuator’s position only. Moreover, by modifying controller dynamics with a saturation function, the control input can be constrained to certain bounds. Finally, simulation results demonstrate the feasibility and effectiveness of the proposed controllers.