Grasping is one of the key tasks for robots. Gripping fragile and complex three-dimensional (3D) objects without applying excessive contact forces has been a challenge for traditional rigid robot grippers. To solve this challenge, soft robotic grippers have been recently proposed for applying small forces and for conforming to complex 3D object shapes passively and easily. However, rigid grippers are still able to exert larger forces, necessary for picking heavy objects. Therefore, in this study, we propose a magnetically switchable soft suction gripper (diameter: 20 mm) to be able to apply both small and large forces. The suction gripper is in its soft state during approach and attachment while it is switched to its rigid state during picking. Such stiffness switching is enabled by filling the soft suction cup with a magnetorheological fluid (MR fluid), which is switched between low-viscosity (soft) and high-viscosity (rigid) states using a strong magnetic field. We characterized the gripper by measuring the force required to pull the gripper from a smooth glass surface. The force was up to 90% larger when the magnetic field was applied (7.1 N vs. 3.8 N). We also demonstrated picking of curved, rough, and wet 3D objects, and thin and delicate films. The proposed stiffness-switchable gripper can also carry heavy objects and still be delicate while handling fragile objects, which is very beneficial for future potential industrial part pick-and-place applications.