This paper investigates an uplink non-orthogonal multiple access (NOMA)-based mobile-edge computing (MEC) network. Our objective is to minimize the total energy consumption of all users including transmission energy and local computation energy subject to computation latency and cloud computation capacity constraints. We first prove that the total energy minimization problem is a convex problem, and it is optimal to transmit with maximal time. Then, we accordingly proposed an iterative algorithm with low complexity, where closed-form solutions are obtained in each step. The proposed algorithm is successfully shown to be globally optimal. Numerical results show that the proposed algorithm achieves better performance than the conventional methods.