Dexterous manipulation is a critical aspect of human capability, enabling interaction with a wide variety of objects. Recent advancements in learning from human demonstrations and teleoperation have enabled progress for robots in such ability. However, these approaches either require complex data collection such as costly human effort for eye-robot contact, or suffer from poor generalization when faced with novel scenarios. To solve both challenges, we propose a framework, DexH2R, that combines human hand motion retargeting with a task-oriented residual action policy, improving task performance by bridging the embodiment gap between human and robotic dexterous hands. Specifically, DexH2R learns the residual policy directly from retargeted primitive actions and task-oriented rewards, eliminating the need for labor-intensive teleoperation systems. Moreover, we incorporate test-time guidance for novel scenarios by taking in desired trajectories of human hands and objects, allowing the dexterous hand to acquire new skills with high generalizability. Extensive experiments in both simulation and real-world environments demonstrate the effectiveness of our work, outperforming prior state-of-the-arts by 40% across various settings.