The Landau damping of the dust ion-acoustic wave (DIAW) propagating in a dusty plasma with nonextensive distributed components is kinetically analyzed. The electron, ion, and dust particles are effectively modeled by nonextensive distributions of Tsallis statistics. For a collisionless plasma with different values of plasma components indices, the general dispersion relation is achieved, and the nonextensivity effects on the frequency, as well as the Landau damping, are studied. We show that for , the preliminary results of the Maxwellian plasma are obtained. The decrease of wave damping is achieved by increasing the coefficient q index and the ion to electron density ratio. The damping rate also increases with a decreasing dust-to-electron temperature ratio.