In this paper, we investigate the qualitative behavior of a fractional-order susceptible-exposed-infected (SEI) model with logistic growth and time delay. In the proposed epidemic model, we assumed that the susceptible individuals grow logistically and introduced time delay in latent infected individuals equation. The basic reproduction number R0 is derived using next generation matrix to study the dynamics of the disease free and endemic equilibrium points of the system. Based on the characteristic equations and conditions of the stability of fractional-order differential systems, local stability of the two equilibrium points is discussed. Furthermore, a suitable Lyapunov function is proposed to investigate the global stability of equilibrium points. The results demonstrated that the fractional-order derivatives enriched the dynamical behavior of the epidemic system. Moreover, in the fractional-order case, the stability region of the equilibrium points increased. The theoretical results are verified by numerical simulations.