This paper investigates the problem of fault tolerant control (FTC) for autonomous underwater vehicles (AUVs) with multiple thrusters in the presence of current disturbances, thruster faults, and modelling uncertainty. This paper focuses on the problems of reducing the energy consumption caused by the chattering of control signals and improving the tracking accuracy of an AUV operating in deep-sea environments. In view of the problem of large energy consumption in some other methods, a fault tolerant control method for multiple-thruster AUVs based on a finite-time extended state observer (FTESO) is proposed. More specifically, a FTESO based on an integral sliding mode surface is designed to estimate the generalized uncertainty compounded using current disturbances, thruster faults, and modelling uncertainty. The fast finite-time uniformly ultimately bounded stability of the proposed FTESO is analyzed. Then, based on the estimated value of FTESO, an FTC method based on non-singular fast terminal sliding mode surfaces is developed for AUVs. The finite-time convergence of the closed-loop control system is proved theoretically. In this design, two different sliding mode surfaces are used to design FTESO and FTC, in order to avoid the appearance of singularities. Moreover, a parameter adjustment method is designed to improve tracking accuracy. Finally, comparative numerical simulations show that the proposed control scheme is effective at reducing energy consumption and improving tracking accuracy.