Mobile edge computing (MEC) can enhance the computing capability of mobile devices, and non-orthogonal multiple access (NOMA) can provide high data rates. Combining these two technologies can effectively benefit the network with spectrum and energy efficiency. In this paper, we investigate the task completion time minimization in NOMA multiuser MEC networks, where multiple users can offload their tasks simultaneously via the same frequency band. We adopt the \emph{partial} offloading, in which each user can partition its computation task into offloading computing and locally computing parts. We aim to minimize the maximum task latency among users by optimizing their tasks partition ratios and offloading transmit power. By considering the energy consumption and transmitted power limitation of each user, the formulated problem is quasi-convex. Thus, a bisection search (BSS) iterative algorithm is proposed to obtain the minimum task completion time. To reduce the complexity of the BSS algorithm and evaluate its optimality, we further derive the closed-form expressions of the optimal task partition ratio and offloading power for two-user NOMA MEC networks based on the analysed results. Simulation results demonstrate the convergence and optimality of the proposed a BSS algorithm and the effectiveness of the proposed optimal derivation.