Robotic cutting of soft materials is critical for applications such as food processing, household automation, and surgical manipulation. As in other areas of robotics, simulators can facilitate controller verification, policy learning, and dataset generation. Moreover, differentiable simulators can enable gradient-based optimization, which is invaluable for calibrating simulation parameters and optimizing controllers. In this work, we present DiSECt: the first differentiable simulator for cutting soft materials. The simulator augments the finite element method (FEM) with a continuous contact model based on signed distance fields (SDF), as well as a continuous damage model that inserts springs on opposite sides of the cutting plane and allows them to weaken until zero stiffness, enabling crack formation. Through various experiments, we evaluate the performance of the simulator. We first show that the simulator can be calibrated to match resultant forces and deformation fields from a state-of-the-art commercial solver and real-world cutting datasets, with generality across cutting velocities and object instances. We then show that Bayesian inference can be performed efficiently by leveraging the differentiability of the simulator, estimating posteriors over hundreds of parameters in a fraction of the time of derivative-free methods. Next, we illustrate that control parameters in the simulation can be optimized to minimize cutting forces via lateral slicing motions. Finally, we conduct experiments on a real robot arm equipped with a slicing knife to infer simulation parameters from force measurements. By optimizing the slicing motion of the knife, we show on fruit cutting scenarios that the average knife force can be reduced by more than 40% compared to a vertical cutting motion. We publish code and additional materials on our project website at https://diff-cutting-sim.github.io.