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Mobile edge computing (MEC) can provide considerable computing capabilities for Internet of Things (IoT) devices, especially for applications with latency sensitive tasks. By applying non-orthogonal multiple access (NOMA) in MEC, multiple users can offload their tasks simultaneously on the same frequency band. In this paper, the minimization problem of task completion time is investigated for the NOMA enabled multi-user MEC networks. We adopt \emph{partial offloading}, in which each user's task can be partitioned, while the formulated problem is quasi-convex. Thus a bisection search (BSS) algorithm is proposed to achieve the minimum task completion time for the multi- user case. To reduce the complexity and evaluate the optimality of the BSS algorithm, we further derive closed- form expressions for the optimal task partition ratio and offloading power for a two-user NOMA-MEC network. Simulations …