This paper investigates the optimal resource allocation of a downlink non-orthogonal multiple access (NOMA) system consisting of one base station and multiple users. Unlike existing short-term NOMA designs that focused on the resource allocation for only the current transmission timeslot, we aim to maximize a long-term network utility by jointly optimizing the data rate control at the network layer and the power allocation among multiple users at the physical layer, subject to practical constraints on both the short-term and long-term power consumptions. To solve this problem, we leverage the recently-developed Lyapunov optimization framework to convert the original long-term optimization problem into a series of online rate control and power allocation problems in each timeslot. The power allocation problem, however, is shown to be non-convex in nature and thus cannot be solved with a standard method. However, we explore two structures of the optimal solution and develop a dynamic programming based power allocation algorithm, which can derive a globally optimal solution, with a polynomial computational complexity. Extensive simulation results are provided to evaluate the performance of the proposed joint rate control and power allocation framework for NOMA systems, which demonstrate that the proposed NOMA design can significantly outperform multiple benchmark schemes, including orthogonal multiple access (OMA) schemes with optimal power allocation and NOMA schemes with non-optimal power allocation, in terms of average throughput and data delay.