The dielectric electro-active polymer (DEAP) actuator, which features large deformation and fast response, has been widely utilized in soft robots. However, the complex hysteresis, creep and quadratic input nonlinearity of the DEAP actuator pose great difficulties and challenges for its controller design. In this article, an inverse compensation-based global fast terminal integral sliding mode control scheme with lumped uncertainty fuzzy estimation is presented for the high-precision tracking control of the DEAP actuator. On the basis of the analytical inverse of a modified Prandtl–Ishlinskii model and a square root module, an inverse compensator is developed to compensate for the hysteresis and the quadratic input nonlinearity of the DEAP actuator. For the system after inverse compensating, a K-observer is adopted to estimate the immeasurable state, and a fuzzy system is used to estimate the lumped uncertainty. Then, a global fast terminal integral sliding mode controller is devised and cascaded in series with the inverse compensator to form a closed-loop control system. The stability and the finite time convergence of the closed-loop control system are proven in theory. Finally, the practical application feasibility and the excellent performance of the developed control scheme are demonstrated via experiments on a testbench for the DEAP actuator.