This work proposes a new method to deal with the stability analysis and stabilization of aperiodic sampled-data control systems subject to input saturation. An impulsive system representation is employed, with a linear flow and a nonlinear jump dynamics, in such a way that the evolution of the system at the sampling instants can be modeled by a difference inclusion defined by two set-valued maps. We show that to ensure the asymptotic stability it is sufficient to verify that a Lyapunov function decreases by a certain amount only at a grid of possible values for the sampling interval, as long as the increase of the function in continuous-time is conveniently bounded. Simulation results compare our approach with other ones.