This paper investigates robust finite-time stability problem of fractional-order systems (FOSs) with time-varying delay and nonlinear perturbation. Based on a generalized Gronwall inequality, a delay-dependent sufficient condition for robust finite-time stability of such systems is established in terms of the Mittag-Leffler function. Compared with the finite-time stability conditions in the literature, our condition is simple and convenient for testing and application. The obtained results are applied to finite-time stability of linear uncertain FOSs and linear non-autonomous FOSs with time-varying delays. A numerical example is given to show the effectiveness of the obtained result.