Millimeter wave (mmWave) communication is a promising technology in future wireless networks because of its wide bandwidths that can achieve high data rates. However, high beam directionality at the transceiver is needed due to the large path loss at mmWave. Therefore, in this paper, we investigate the beam alignment and power allocation problem in a nonorthogonal multiple access (NOMA) mmWave system. Different from the traditional beam alignment problem, we consider the NOMA scheme during the beam alignment phase when two users are at the same or close angle direction from the base station. Next, we formulate an optimization problem of joint beamwidth selection and power allocation to maximize the sum rate, where the quality of service (QoS) of the users and total power constraints are imposed. Since it is difficult to directly solve the formulated problem, we start by fixing the beamwidth. Next, we transform the power allocation optimization problem into a convex one, and a closed-form solution is derived. In addition, a one-dimensional search algorithm is used to find the optimal beamwidth. Finally, simulation results are conducted to compare the performance of the proposed NOMA-based beam alignment and power allocation scheme with that of the conventional OMA scheme.