Human-robot object handovers are essential for robots to effectively serve human needs in various domains of human–robot interaction and collaboration, yet remain a significant challenge. Remarkable progress has been made by parallel-jaw gripper robots in grasp generation and motion planning for handovers, while few studies address this issue using anthropomorphic hands, necessitating the ability to handle higher collision probabilities and lower approaching space under occluded situations. In this article, we present a reactive human-to-robot dexterous handover framework for anthropomorphic hands. The closed-loop framework employs an effective collision detection and grasp selection approach to ensure safe and smooth motion in unstructured environments. We implement a handover system using a UR5 robot arm and a Schunk SVH Hand based on the presented framework, which can react to human motion during the handover process, generalize to diverse objects with different 6-DoF poses, and execute suitable grasp configurations. The generalizability, reliability, and efficiency of our method are demonstrated through the handover of 30 novel objects, a system ablation study for submodule evaluation, and a user study assessment involving eight participants.