This paper designs a novel non-orthogonal multiple access (NOMA) unicast–multicast system, where a number of unicast users (those who require different messages) and a group of multicast users (those who require identical message) share the same time/space/frequency resource. For the designed NOMA unicast–multicast system, an efficient two-phase cooperation strategy is proposed to improve the reliability of all users. In the first phase, the base station (BS) broadcasts a superposed message consisting of all users’ information. In the second phase, a multicast user is selected to forward the information intended by unsuccessfully decoded unicast and/or multicast users. Moreover, the multicast user selection is investigated under two different power allocation (PA) approaches: 1) fixed PA (FPA), in which the PA coefficients for both the phases are predetermined, and 2) dynamic PA (DPA), in which the PA coefficients for the first phase are predetermined, while the PA coefficients for the second phase are dynamically determined based on instantaneous channel information. Under the FPA approach, a best user selection (BUS) scheme (called F-BUS) is proposed to minimize the outage probability. Under the DPA approach, the local optimal PA coefficients for the second phase are derived in closed form first. Based on the derived PA coefficients, a BUS scheme (called D-BUS) is then proposed for outage probability minimization. To verify the reliability of the proposed cooperation strategy with employing the BUS schemes, we theoretically analyze the outage probability as well as diversity orders. It is shown that the proposed cooperation strategy achieves diversity orders equal to the number of multicast users, indicating that the inherent diversity orders offered by the multicast users are fully exploited. Finally, simulation results are presented to validate the theoretical results and demonstrate the advantages of the proposed cooperation strategy and the BUS schemes.