This paper considers the problem of finite-time disturbance observer (FTDO) design and the problem of FTDO-based finite-time control (FTC) for systems subject to nonvanishing disturbances. First of all, based on the homogeneous domination approach and saturation technique, a continuous FTDO is proposed. Then, by using the proposed FTDO design method, an FTDO is constructed to estimate the disturbances that exist in a rigid spacecraft system. Furthermore, a composite controller consisting of a baseline FTC law and a feedforward compensation term based on the estimation of FTDO is constructed for the rigid spacecraft system. It is shown that under the proposed composite controller the rigid spacecraft can track the desired attitude trajectory in a finite time. Simulation results are included to demonstrate the effectiveness of the proposed control approach.