High dynamic performance of a deep-sea electric oil-filled joint actuator is an important premise to guarantee the working performance of an electric underwater manipulator. However, the unfavorable factors (i.e., extremely high water pressure, near freezing temperature) brought by the deep-sea working environment seriously affect the characteristic and dynamic performance of the electric oil-filled joint actuator, which mainly includes oil stirring viscos loss, output shaft dynamic seal loss, and core loss. In this paper, a novel observer-based robust control method named prescribed performance non-singular fast-terminal sliding-mode control (PP-NFTSMC-ESO) was synthesized for improving the dynamic performance of a deep-sea electric oil-filled joint actuator. The extended state observer (ESO) was employed to observe the unmeasured joint velocity signal and estimate the lumped uncertainties, while the prescribed performance function (PPF) was applied to constrain the instantaneous and steady-state performance of the trajectory-tracking error. The robust NFTSMC control method was then established by integrating the function of ESO and PPF through backstepping methodology. The stability of the proposed PP-NFTSMC-ESO strategy was analyzed and proved by the Lyapunov’s stability theory. It was proven that under the proposed controller, all the closed-loop signals are bounded and the trajectory tracking errors will converge to a small neighborhood of the origin with appropriate design parameters. The effectiveness of the proposed control scheme was illustrated by comparative simulation studies.