While suction cups prevail as common gripping tools for a wide range of real-world parts and surfaces, they often fail to seal the contact interface when engaging with irregular shapes and textured surfaces. In this work, the authors propose a suction-based soft robotic gripper where suction is created inside a self-sealing, highly conformable, and thin flat elastic membrane contacting a given part surface. Such a soft gripper can self-adapt the size of its effective suction area with respect to the applied load. The elastomeric membrane covering the edge of the soft gripper can develop an air-tight self-sealing with parts even smaller than the gripper diameter. Such gripper shows 4 times higher adhesion than the one without the membrane on various textured surfaces. The two major advantages, underactuated self-adaptability and enhanced suction performance allow the membrane-based suction mechanism to grip various three-dimensional (3D) geometries and delicate parts, such as egg, lime, apple, and even hydrogels without noticeable damage, which can have not been gripped with the previous adhesive microstructures-based and active suction-based soft grippers. The structural and material simplicity of the proposed soft gripper design can have broad use in diverse fields, such as digital manufacturing, robotic manipulation, transfer printing, and medical gripping.