In this paper, the finite-time trajectory tracking control problem of a flexible link manipulator (FLM) system with unknown parameters is investigated in joint space. An adaptive nonsingular terminal sliding mode (ANTSM) controller based on an extended state observer (ESO) is proposed to ensure that the tracking error converges to a small neighborhood of zero. Firstly, an adaptive ESO is introduced, which is used to estimate unknown system parameters, including the unknown rigid-flexible coupling coefficient and the unknown control gain. Then, an ESO based on the parameters estimated above is proposed to estimate the unmodeled dynamics and external disturbances of the system in real-time. Finally, an ANTSM controller based on ESO is studied to achieve finite-time trajectory tracking control and vibration suppression without any prior knowledge of system uncertainties and external disturbances. The finite-time convergence ability of the closed-loop system is proved by Lyapunov theory. Experimental results on the Quanser Flexible Link System verify the effectiveness of the proposed controller.