This article proposes a fractional-order sliding-mode control (FOSMC) method based on the super-twisting algorithm to compensate for a general form of mismatch uncertainties in a class of nonlinear systems. First, a new super-twisting algorithm-based fractional-order sliding-mode virtual control law is constructed to ensure that the state trajectory can converge to the equilibrium point and avoid the singular problem caused by the system state differential with fractional powers. Furthermore, the switching term in the virtual control law is softened by a $(2\alpha +1)$-order integrator. Unlike the traditional boundary layer method, the fractional-order switching term in the designed control law has continuous derivatives so that a continuous actual control signal can be obtained without sacrificing control accuracy. With continuous control, the nonlinear system can respond quickly with high precision. Finally, simulation and experimental results demonstrate that the proposed control method exhibits excellent control performance in nonlinear systems with matched and mismatched uncertainties.