This paper presents a novel nonlinear backstepping control law for continuous, low-thrust station-keeping in the Earth-Moon system. Quasi-periodic libration point orbits are targeted under a high-fidelity model of the dynamics. Almost global uniform exponential stability guarantees are attained, as shown through Lyapunov's stability theory. Saturation of the actuators is formally included in the controller design, such that these guarantees hold even in the event of saturation. The relationship between saturation threshold, control gains, and deviation is studied and an optimal procedure for gain selection is discussed. The control solution is tested numerically through a Monte Carlo analysis over representative application cases, subject to operational errors, constraints, and external perturbations. Station-keeping under actuation saturation is validated considering a conservative threshold for typical electric propulsion systems.