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