This paper presents a new global terminal sliding mode control (GTSMC) methodology for the simultaneous positioning and vibration suppression of slewing a flexible-link manipulator. The proposed control scheme is realized using only a joint actuator without an extra actuator (for example, the widely used piezoelectric zirconate titanate), and the non-minimum phase vibration control problem is solved. Firstly, based on the differential geometry method, the initial dynamic system is decomposed into two subsystems, namely an input-output subsystem and a zero-dynamics subsystem. Secondly, a continuous GTSMC strategy without chattering phenomenon is designed for the input-output subsystem, and the limited convergence time is deduced. Moreover, the eigenvalues of the zero-dynamics subsystem can be pointed by setting proper controller parameters for Lyapunov asymptotical stability. Finally, simulation and experimental results demonstrated the efficacy and feasibility of the proposed control methodology.