This paper presents a strategy meant to allow anthropomorphic robot hands to grasp objects of unknown shape in imitation of the human grasping reflex. A 10-month-old baby may bend his/her thumb and four fingers in an attempt to grasp an object when it comes in contact with the palm. After grasping it, if the object is attempted to be removed from the baby's hand, the baby holds the object more strongly. In addition, the palm of the hand often makes initial contact with the object very lightly. This reaction is called the grasping reflex. In the proposed grasping strategy, each joint of the thumb and the fingers of a robot hand is controlled independently using the contact force affecting its adjacent fingertip side-link to imitate the grasping reflex. By setting a suitable contact force, the robot can grasp an object by both the fingertips as well as an enveloping grasp involving the palm at a uniform grasping force. Experimental results regarding the grasping of three-dimensional objects of unknown shape by an anthropomorphic robot hand called the Gifu Hand III are shown.