Soft anthropomorphic robotic grippers are attractive because of their inherent compliance, allowing them to adapt to the shape of grasped objects and the overload protection needed for safe human–robot interaction or gripping delicate objects with sophisticated control. The anthropomorphic design allows the gripper to benefit from the biological evolution of the human hand to create a multi-functional robotic end effector. Entirely soft grippers could be more efficient because they yield under high loads. A trending solution is a hybrid gripper combining soft and rigid elements. This work describes a prototype of an anthropomorphic, underactuated five-finger gripper with a direct pneumatic drive from soft bending actuators and an integrated resistive tactile sensor array. It is a hybrid construction with soft robotic structures and rigid skeletal elements, which reinforce the body, focus the direction of the actuator’s movement, and make the finger joints follow the forward kinematics. The hand is equipped with a resistive tactile dielectric elastomer sensor array that directly triggers the hand’s actuation in the sense of reflexes. The hand can execute precision grips with two and three fingers, as well as lateral grip and strong grip types. The softness of the actuation allows the finger to adapt to the shape of the objects.