A robust finite-time control (FTC) framework using continuous terminal sliding-mode control (SMC) and high-order sliding-mode observer (HOSMO) is discussed to realize the trajectory tracking of flexible-joint robots in this article. Control performances of the robots always suffer from unknown matched and mismatched time-varying disturbances. Traditional SMC exists with a chattering phenomenon and cannot cope with mismatched time-varying disturbances due to its inherent structure property. For this reason, two HOSMOs are devised to estimate the time-varying disturbances on the link and motor side, respectively. Then, by fusing the states and disturbance estimations into a novel terminal sliding-mode surface, a continuous robust FTC scheme is developed. The proposed control strategy can not only handle both matched and mismatched time-varying disturbances but also obtain a finite-time convergence performance. The rigorous finite-time stability analysis of the closed-loop system under the proposed control method is guaranteed. The results are illustrated to verify the effectiveness and robustness of the proposed design approach.