A novel anti-saturation robust control algorithm (NARC) is presented for flexible air-breathing hypersonic vehicle (FAHV) with actuator saturation, including two controllers designed for velocity and height subsystem respectively. Firstly, an anti-saturation finite-time dynamic inversion controller is designed for velocity subsystem, in which an anti-saturation fixed-time compensator (ASFC) is proposed to ensure the stability of saturated system and make it exit saturated region faster. Compared with conventional anti-saturation compensator, the auxiliary variable of ASFC can converge with faster speed and higher precision when actuator is not saturated, which avoids the impact on original system. Secondly, an anti-saturation robust command filtered backstepping controller is designed for height subsystem, combining backstepping control, ASFC and a novel fixed-time filter (FTF). Compared with low pass filter, the FTF proposed can track input signal with faster response speed and higher precision without the need to select a smaller time constant, so as to avoid introducing high-frequency noise. Meanwhile, convergence domain of height subsystem can be reduced as well. Ultimately, simulations on FAHV with actuator constraints, parametric uncertainties and external disturbances are performed using the NARC and conventional anti-saturation controller respectively to demonstrate the superiority of NARC.