In this letter, we investigate covert communications in an active reconfigure intelligent surface (RIS) aided multiple-input single-output non-orthogonal multiple access system. To solve the covert rate maximization problem of joint optimizing the transmit power allocation, transmit beamforming, and reflection beamforming, we decouple the original non-convex problem into three sub-problems and propose an alternative optimization (AO) algorithm to solve the sub-problems. To tackle the rank-1 constrained fractional programming problem of the reflection beamforming optimization, a penalized Dinkelbach algorithm is proposed, while the rank-1 constrained reflection beamforming optimization problem is solved with the aid of a penalty-term. Moreover, the optimal transmit beamforming is derived in a determined way. Numerical results clarify the significantly boosted covert rate achieved by the proposed AO algorithm over that of the passive RIS counterpart.