We consider a new class of differential quasivariational inequalities, i.e. a nonlinear system that couples a differential equation with a time-dependent quasivariational inequality, both defined on abstract Banach spaces. We state and prove a general fixed principle that provides the existence and the uniqueness of the solution of the system. Then we consider a relevant particular setting for which our abstract result holds. We proceed with two examples that arise in Contact Mechanics. For each example, we describe the physical setting, the mathematical model and the assumption on the data. Then we state the variational formulation of each model, which is in the form of a differential quasivariational inequality. Finally, we apply our abstract results to provide the unique weak solvability of the corresponding contact problems.