In this paper, we first employ non-orthogonal multiple access (NOMA) in a cognitive radio (CR) based mobile edge computing (MEC) network to reduce the system delay. In particular, each secondary user has two computation tasks which can be offloaded to the MEC servers via NOMA, while the transmissions of different secondary users are orthogonal by exploiting different spectrum bands. Then, we investigate the system delay minimization problem by jointly optimizing the offloading policy, the transmit power for offloading, the computing capabilities for local secondary users and MEC servers. Since the formulated problem is non-convex and difficult to be solved straightforwardly, we first analyze its characteristics and decompose it into four sub-problems, which can be solved independently. After that, we propose a low-complexity algorithm to obtain the sub-optimal solution. Simulation results prove the superiority of the proposed scheme in terms of the system delay compared with the equal allocation scheme. Our proposed scheme can achieve a sub-optimal solution which is very close to the optimal solution with low complexity compared with the Branch and Bound method. INDEX TERMS NOMA, cognitive radio, mobile edge computing, resource allocation.