In this paper, we propose a downlink multiple-input single-output (MISO) transmission scheme, which is assisted by an intelligent reflecting surface (IRS) consisting of a large number of passive reflecting elements. In the literature, it has been proved that nonorthogonal multiple access (NOMA) can achieve the same performance as computationally complex dirty paper coding, where the quasi-degradation condition is satisfied, conditioned on the users’ channels fall in the quasi-degradation region. However, in a conventional communication scenario, it is difficult to guarantee the quasi-degradation, because the channels are determined by the propagation environments and cannot be reconfigured. To overcome this difficulty, we focus on an IRS-assisted MISO NOMA system, where the wireless channels can be effectively tuned. We optimize the beamforming vectors and the IRS phase shift matrix for minimizing transmission power. Furthermore, we propose an improved quasi-degradation condition by using IRS, which can ensure that NOMA achieves the capacity region with high possibility. For a comparison, we study zero-forcing beamforming (ZFBF) as well, where the beamforming vectors and the IRS phase shift matrix are also jointly optimized. Comparing NOMA with ZFBF, it is shown that, with the same IRS phase shift matrix and the improved quasi-degradation condition, NOMA always outperforms ZFBF. At the same time, we identify the condition under which ZFBF outperforms NOMA, which motivates the proposed hybrid NOMA transmission. Simulation results show that the proposed IRS-assisted MISO system outperforms the MISO case without IRS, and the hybrid NOMA transmission scheme always achieves better performance than orthogonal multiple access.