Energy-efficient communication is one of the most pertaining issues in realizing mobile edge computing (MEC) networks. How to effectively use non-orthogonal multiple access (NOMA) to realize energy efficient MEC networks has been of high interest and calls for in-depth study. In particular, there has been yet no work to study the energy efficient communication scheme from system perspective in this area. In this work, we focus on the design of a system-centric computation energy efficiency (EE) scheme in a NOMA-based MEC network. The system computation EE is maximized by jointly optimizing the computing frequencies of the MEC server and the local users, the offloading time, the transmit powers of local users, as well as the execution time for the local users and the MEC server. Employing the fractional programming theory and introducing auxiliary variables, we develop an efficient Dinkelbach-based iterative scheme to tackle the formulated non-linear fractional programming problem. For obtaining instrumental insights, we further derive closed-form expressions for the partial solutions based on Karush-Kuhn-Tucker conditions. Simulation results are provided to verify the quick convergence of the proposed iterative algorithm, to obtain insights, as well as to demonstrate the superiority of the proposed resource allocation scheme in terms of the system computation EE by comparing it with benchmark schemes.