We propose a dynamical system-based approach for solving robust optimization problems. The well-known continuous-time dynamical system for solving deterministic optimization problems arises in the form of primal-dual gradient dynamics where the vector field is derived as the gradient of the Lagrangian. The new continuous-time dynamical system we introduce for solving robust optimization problems differs from the primal-dual dynamics in the sense that the vector field is not derived as the gradient of the Lagrangian function. We call this new dynamical system as saddle point dynamics. In the saddle point dynamics, the uncertain variable arises as a dynamical state. For a general class of robust optimization problem, where the cost function is convex in decision variable and concave in uncertain variable, we show that the robust optimal solution can be recovered as a globally asymptotically stable equilibrium point of the saddle point dynamical system. Simulation results are presented to demonstrate the capability of this new dynamical system to solve various robust optimization problems. We also compare our proposed approach with existing methods based on robust counterpart and scenario-based random sampling.