In this article, we investigate the cellular assisted mobile edge computing (MEC) via non-orthogonal multiple access (NOMA), where a group of edge-computing users (EUs) exploit NOMA to simultaneously offload their computation-workloads to an edge-server (ES), and conventional cellular-user (CU) allows the EUs to reuse its authorized frequency channel for the NOMA-transmission. We firstly characterize the transmit-powers of the CU and EUs and then formulate a joint optimization of the EUs' offloaded computation-workloads, offloading-duration (i.e., how long for reusing the CU's channel), as well as the ES's computation-resource allocations (for processing different EUs' offloaded workloads), with the objective of minimizing the overall-latency in completing all EUs' computation-requirements, subject to the CU's and EUs' limited power and energy capacities as well as the ES's limited computation-resource capacity. Despite the strict non-convexity of the formulated joint optimization problem, we propose an efficient algorithm to compute the optimal offloading solution. With the optimal offloading solution for the scenario of one CU, we further investigate the scenario of multiple CUs and investigate the optimal pairing of the EUs for reusing different CUs' channels for computation-offloading. Taking into account the coupling effect due to the ES's limited computation-resource, we formulate a joint optimization of the EU-pairing and the ES's capacity allocation of the computation-resource for accommodating different EU-pairs. Despite the difficulty due to the mixed binary and non-linear non-convex programming of the formulated problem, we propose an efficient layered algorithm for solving the problem. Numerical results are provided to validate the accuracy of our proposed algorithms. We also show the performance advantage of our NOMA-assisted offloading in comparison with conventional orthogonal multiple access (OMA) based computation offloading.