This paper introduces an adaptive control method for finite-time modified function projective lag synchronization of uncertain hyperchaotic systems. Based upon novel nonsingular terminal sliding mode surfaces and the adaptive super-twisting algorithm, a controller is proposed to provide robustness, high precision and fast and finite-time modified function projective lag synchronization without the knowledge of the upper bound of uncertainties and unknown external disturbances. In addition, chattering is significantly attenuated due to the inherited continuity of the proposed controller. The global stability and finite-time convergence are rigorously proven. Numerical simulation is presented to demonstrate the effectiveness and feasibility of the proposed strategy and to verify the theoretical results.