This paper presents a dynamic feedback framework for the control of time-delay cascade systems with unstable zero dynamics. Both dynamic state and output feedback control strategies are studied. Recognizing the difficulty of controlling time-delay systems via static feedback, we develop a systematic method, by taking advantage of dynamic feedback, for the design of delay-free, dynamic state and output feedback compensators that achieve global state regulation with stability. The controlled plants under consideration not only cover time-delay nonlinear systems in the normal form, but also include a class of time-delay cascade systems beyond the normal form. In the case of state feedback, the zero dynamics are not required to be "minimum phase" but satisfy certain regularity conditions. In the output feedback case, appropriate conditions are characterized for the zero dynamics so that the existence of a dynamic output controller is ensured. A key feature of the proposed control strategies is the utilization of dynamic gains to counteract the effect of time-delay nonlinearities and unstable zero dynamics.