To improve the convergence rate of standard sliding mode control whose convergence rate is exponential, a robust finite-time control algorithm for satellite attitude stabilization under control torque saturation is presented. A three-stage structure finite time sliding mode with better convergence rate and steady accuracy is developed based on the Lyapunov method, and the issue that severe descent of angular velocity leads to convergence rate decline of attitude quaternion is avoided. The singularity issue is solved by using the property of Euler axis. By limiting the proportional term in control law, the control torque constraint is added into the control law. The known disturbance torque and inertia matrix uncertainty is solved by introducing the sign function. The finite-time stability is proved by a Lyapunov function, the convergence time estimation is given, and the property that the proposed controller could largely improve the convergence rate and maintain the high accuracy at steady stage is demonstrated by theoretical analysis and simulation results. This paper proves that planning angular velocity is the key to improve system performance, and the convergence rate and the system robustness could be improved by properly designing sliding mode and desired angular velocity.