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 soft gripper can self-adapt the size of its effective suction area with respect to the applied load. The elastomeric membrane covering 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 a broad use in diverse fields, such as digital manufacturing, robotic manipulation, transfer printing, and medical gripping.