Due to the limited computation resources of mobile devices in cognitive radio networks, the secondary users in the network can suffer from long executing time, which is not acceptable for latency‐sensitive and computation‐intensive tasks. To tackle this issue, this paper proposes to reduce the task computing latency for secondary networks by offloading the tasks to edge servers through leveraging mobile edge computing (MEC) that is emerging as a promising technology to augment the computation capacity of mobile devices. Specifically, under the conditions that the interference caused by secondary users is tolerable to primary user and within the available computation resources of the MEC server, the primary user and secondary users both can offload tasks to the MEC server through nonorthogonal multiple access. Thus, we jointly formulate the offloading decision and power control as an optimization problem, aiming at minimizing the overall computing latency for secondary networks. To overcome the computational complexity caused by the nonconvexity of the original problem, we transform the original problem to a solvable problem and decouple the transformed problem into the separate offloading decision and power control. An iterative algorithm is proposed based on block coordinate decent method to achieve the near‐optimal solution. Simulation results show that under the same parameters, such as the number of primary users, maximum transmit power, computational capability of the MEC server and the computational capability of the secondary users, the proposed NOMA‐enabled computation offloading scheme can effectively reduce the overall computing latency for the secondary network and improve the percentage of offloading secondary users than those of OMA‐enabled.