In this paper, we investigate integrated sensing and communication (ISAC) in high-mobility systems with the aid of an intelligent reflecting surface (IRS). To exploit the benefits of Delay-Doppler (DD) spread caused by high mobility, orthogonal time frequency space (OTFS)-based frame structure and transmission framework are proposed. {In such a framework,} we first design a low-complexity ratio-based sensing algorithm for estimating the velocity of mobile user. Then, we analyze the performance of sensing and communication in terms of achievable mean square error (MSE) and achievable rate, respectively, and reveal the impact of key parameters. Next, with the derived performance expressions, we jointly optimize the phase shift matrix of IRS and the receive combining vector at the base station (BS) to improve the overall performance of integrated sensing and communication. Finally, extensive simulation results confirm the effectiveness of the proposed algorithms in high-mobility systems.