Over the past few decades, quasi-variational inequalities (QVIs) have gained significant interest among researchers as an extension of the well-known variational inequalities (VIs). This extension is characterized by the dependence of the constraint set on the current value of the point. The interest in QVIs is attributed to their capability to model various real-world problems in game theory, economics, optimization and traffic equilibrium. This paper proposes a novel dynamical system (DS) for solving a QVI. We prove an existence and uniqueness result for the considered problem using the Banach contraction principle. Then, we establish the global exponential stability (GES) of the proposed DS under some reasonable conditions on the parameters. Finally, the DS is discretized to an iterative method through explicit discretization, and its convergence to the solution of the considered QVI is established.