In this paper, we present a dynamic resource management scheme for delay-aware applications in two-tier small-cell networks (SCNs). We propose the scheme of joint rate control at the transport layer and resource allocation at the physical layer to manage the cross-tier interference. The joint rate control and resource allocation scheme is designed to maximize the time-averaged sum capacity of small cell users in the SCN subject to each small cell user's delay constraint and an interference constraint imposed by the macrocell. By using the Lyapunov optimization technique, we develop a delay-guaranteed capacity optimal algorithm (DCOA) to obtain the optimal rate control and resource allocation decisions. We show that without prior knowledge of the data arrivals and channel statistics, DCOA achieves a capacity of SCN that can arbitrarily approach the optimal capacity achieved by the algorithm with the complete knowledge of data arrivals and channel statistics. Simulations results confirm the theoretical analysis on the performance of DCOA and also show the adaptiveness of DCOA.