This work proposes a novel finite-time backstepping framework combined with an auxiliary input-saturation compensator, and applies it to tackle the trajectory tracking problem of quadrotors with disturbances and input constraints. Finite-time convergence property is guaranteed by introducing the fractional power functions of tracking errors and the finite-time filters of target commands for the next step. To avoid the singularity of rotation motion, quaternion-based attitude control is adopted, which target command is obtained from the finite-time filter of desired thrust direction. Additionally, finite-time disturbance observers based on multivariable super-twisting algorithm are employed to estimate the lumped perturbations, and then we directly counteract them. To handle the adverse effect of input saturation, a novel auxiliary system is developed to provide fast desaturation when input saturation occurs. The singularity of the auxiliary dynamics is avoided by the cubic representation of the auxiliary variables. A rigorous proof of finite-time stability of the closed-loop system is derived by the Lyapunov theory, despite the presence of input saturation and disturbances. Finally, several comparative simulations and experimental results demonstrate the effectiveness and superiority of the proposed method.