Based on an integral backstepping approach, a trajectory-tracking control algorithm is proposed for an underactuated unmanned marine vehicle (UMV) sailing in the presence of ocean-current disturbance. Taking into consideration the UMV model's fore/aft asymmetry, a nonlinear three-degree-of-freedom (3DOF) underactuated dynamic model is established for the horizontal plane. First, trajectory-tracking differences between controllers designed based on symmetric and asymmetric models of the UMV are discussed. In order to explicitly study the effect of ocean-current interference on the trajectory-tracking controller, the ocean current is integrated into the kinematic and dynamic models of the UMV. Detailed descriptions of distinct trajectory-tracking control performances in the presence of different ocean-current velocities and direction angles are presented. The well-known persistent exciting (PE) condition is completely released in the designed trajectory-tracking controller. A mild integral item of trajectory tracking error is merged into the control law, and global stability analysis of the UMV system is carried out using Lyapunov theory and Barbalat's Lemma. Simulation experiments in the semi-physical simulation platform are implemented to confirm the effectiveness and superiority of the excogitated control algorithm.