Millimeter wave (mmWave) communications are evolving as a promising technology to meet the ever increasing data rate requirements. However, high directivity and severe path loss make it vulnerable to blockages, which could be frequent in indoor or urban environments. To address this issue, intelligent reflecting surfaces (IRSs) are introduced to provide additional adjustable reflected paths. Most prior works assume that elements of IRSs have an infinite phase resolution, which is difficult to be realized in practical systems. In this paper, IRSs with low-resolution phase shifters are considered. We aim to maximize the receive signal power at the user by jointly optimizing discrete phase shifts of IRSs and the transmit beamforming vector at the base station for mmWave downlink systems. An analytical near-optimal solution is developed by exploiting some important characteristics of mmWave channels. Our theoretical analysis reveals that low-resolution phase shifters can still achieve a receive signal power that increases quadratically with the number of reflecting elements. Simulation results are provided to corroborate our analysis and show the effectiveness of the proposed solution.