Many objects such as tools and household items can be used only if grasped in a very specific way - grasped functionally. Often, a direct functional grasp is not possible, though. We propose a method for learning a dexterous pre-grasp manipulation policy to achieve human-like functional grasps using deep reinforcement learning. We introduce a dense multi-component reward function that enables learning a single policy, capable of dexterous pre-grasp manipulation of novel instances of several known object categories with an anthropomorphic hand. The policy is learned purely by means of reinforcement learning from scratch, without any expert demonstrations, and implicitly learns to reposition and reorient objects of complex shapes to achieve given functional grasps. Learning is done on a single GPU in less than three hours.