This paper is devoted to the study of a class of sweeping processes with history-dependent operators. A well-posedness result is obtained, including the existence, uniqueness, and stability of the solution. Our approach is based on the variable time step-length discrete approximation method combined with a fixed point principle for history-dependent operators. Then, a quasi-static frictional contact problem for viscoelastic materials with unilateral constraints in velocity is considered. The abstract result is applied in the study of this problem in order to provide its unique weak solvability as well as the continuous dependence of the solution with respect to the initial data.