Pneumatic soft robots are promising in diverse applications while they typically require additional electronics or components for pressure control. Fusing pneumatic actuation and control capabilities into a simple soft module remains challenging. Here, we present a class of bistable fabric mechanisms (BFMs) that merge soft bistable actuators and valves for electronics-free autonomous robots. The BFMs comprise two bonding fabric chambers with embedded tubes, where the straightening of one chamber compels the other to buckle for the bistability of the structure and the switching of the tube kinking. Our BFMs can facilitate fast bending actuation (more than 1166° s), on/off and continuous pressure regulation, pneumatic logic computations, and autonomous oscillating actuation (up to 4.6 Hz). We further demonstrate the capabilities of BFMs for diverse robotic applications powered by one constant-pressure air supply: a soft gripper for dynamic grasping and a soft crawler for autonomous jumping. Our BFM development showcases unique features and huge potential in advancing entirely soft, electronics-free autonomous robots.