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This paper proposes a device-to-device (D2D) enabling cellular full-duplex (FD) cooperative (DFC) protocol using non-orthogonal multiple access (NOMA) called DFC-NOMA, where an FD relay acting D2D transmitter assists in relaying a NOMA far user’s signal and transmits a D2D receiver’s signal simultaneously. The ergodic capacity, outage probability, and diversity order of DFC-NOMA are theoretically investigated under the assumption of both perfect and imperfect interference cancellation. The theoretical analysis is then validated by simulations. Both analysis and simulation results demonstrate the performance gain of DFC-NOMA over conventional FD cooperative NOMA and existing D2D aided FD NOMA.