In hybrid device-to-device (D2D) and cellular networks, the cross interference between different communication systems severely limits the system performance. Fortunately, non-orthogonal multiple access (NOMA)-based coordinated direct and relay transmission (CDRT) can fully exploit the inherent property of NOMA to minimize such interference, which provides a potential strategy for hybrid networks to enhance the spectral efficiency (SE) and extend the cell coverage. Based on this idea and the superior SE requirements in future wireless networks, a D2D communication assisted uplink CDRT using NOMA is proposed, in which a base station (BS) directly communicates with a cell-center user (CCU) while communicating with a cell-edge user (CEU) with the aid of a relay. Particularly, uplink transmission of CCU and D2D communication from CCU to CEU are jointly designed according to NOMA principle in the cooperative phase. To verify the system performance of the proposed scheme, closed-form expressions of the system ergodic sum rate (SR), outage performance and outage throughput are derived, respectively. Numerical results illustrate that the proposed scheme achieves superior system ergodic SR as well as outage throughput with slight outage performance loss of uplink transmission, compared with conventional NOMA-based uplink CDRT.