In this paper, we propose a beamwidth control-based non-orthogonal multiple access (NOMA) scheme for hybrid millimeter wave (mmWave) communication systems. In particular, the proposed scheme allows multiple users in one NOMA group to share the same radio frequency chain and analog beam for superposition transmission. To overcome the physical limit of the narrow analog beam, a beamwidth control approach is proposed to widen the analog beamwidth to facilitate the formation of NOMA groups. Then, we characterize the main lobe power loss associated with the proposed beamwidth control and derive the asymptotically optimal analog beamformer to maximize the system sum-rate in the large number of antennas regime. The system sum-rate gain of the proposed beamwidth control-based NOMA scheme compared to a baseline scheme adopting time division multiple access (TDMA) is analyzed. Simulation results verify the accuracy of our performance analysis and unveil the importance of beamwidth control for practical mmWave NOMA systems.