The flexible joints and unknown model dynamics in the flexible joint manipulators lead to difficulties in the controller design. Traditional adaptive control methods typically presume the persistent excitation (PE) condition to estimate the unknown model parameters, but this condition is difficult to satisfy in practical applications. To address this issue, a prescribed-time adaptive control under the finite excitation (FE) is proposed. First, by applying multiple low-pass filters on the system measurements, this method stores the excited information and thus relaxes the PE condition. Then, a prescribed-time convergence adaptive control is designe based on the backstepping method and dynamic surface control (DSC). The proposed method with time-varying gains ensures that the tracking error and parameter estimation error converge to an arbitrarily small set within the prescribed-time. Simulation and experimental results demonstrate that the proposed method achieves high-precision tracking control and enhances the robustness under the FE conditions for the flexible joint manipulators.