In recent years, the 3rd generation partnership project (3GPP) has approved the narrowband Internet of Things (NB-IoT) system to support the low-data-rate machine- type communications. With the rapid development of NB- IoT technology, the NB-IoT traffic volumes have been experiencing the unprecedented growth. To this end, non-orthogonal multiple access (NOMA) and mobile edge computing (MEC) have been proposed as promising technologies for the NB-IoT system. In this paper, our goal is to minimize the total energy consumption subject to the computation capacity and execution latency limits by jointly optimizing the transmit power, computation resource allocation, and successive interference cancellation (SIC) ordering. Considering the NP-hardness of the joint optimization problem, we obtain the optimal solution in the coupled steps: the resource allocation optimization and the combinatorial SIC ordering optimization. By exploiting the convex nature of the resource allocation optimization problem, we obtain the optimal transmit power and computation resource allocation based on the KKT conditions and the idea of gradient descent method when fixing the SIC ordering. Considering the combinatorial optimization of SIC ordering, we further propose a tabu search based SIC ordering algorithm on the basis of the proposed resource allocation optimization algorithm. Finally, simulation results demonstrate that the proposed joint optimization of transmit power, computation resource allocation, and SIC ordering in the context of NOMA can effectively reduce the total energy consumption of MEC- aware NB-IoT system, in comparison with the frequency- division multiple access technique.