The orbital angular momentum (OAM)-based communication systems may face severe transmission problems when the transmit and receive uniform circular array pairs are blocked. In this paper, a promising technique named intelligent reflecting surface (IRS) is proposed to help alleviate blockages and provide alternative line-of-sight links. To maximize the achievable rate of the IRS-assisted OAM communication systems, we optimize its transmit power allocation along with the IRS's reflecting phase shifts, and propose an alternative optimization-based algorithm to solve the resulting optimization problem with coupled variables and non-convex structure. Specifically, the proposed algorithm obtains a closed-form solution to the transmit power allocation by applying the majorization-minimization and $\ell _1$-ball projection approaches, and obtain the locally optimal solution to the IRS's reflecting phase shifts by applying the weighted minimum mean square error-based fixed point iteration approach. Simulation results demonstrate the superiority of our proposed algorithm over existing baseline algorithms and also show its robust stability to the oblique angle errors.