The linear characteristics of the unstable mode of ion-acoustic waves are examined in an electrostatic electron-ion plasma composed of streaming hot electrons,non-streaming cold electrons and dynamical positive ions.The plasma under consideration is modeled by using a non-gyrotropic nonextensive q-distribution function in which the free energy source for wave excitation is provided by the relative directed motion of streaming hot electrons with respect to the other plasma species.In the frame work of kinetic model,a linearized set of Vlasov-Poisson\'s equations are solved to obtain the analytical expressions for dispersion relation and Landau damping rate.The threshold condition for the unstable ion-acoustic wave is derived to assess the stability of the wave in the presence of nonextensive effects.Growth in the wave spectrum and nontrivial effects of q-nonextensive parameter on the ion-acoustic waves can be of interest for the readers in the regions of Saturns\' s magnetosphere.