This paper studies a distributed algorithm for constrained consensus optimization that is obtained by fusing the Arrow-Hurwicz-Uzawa primal-dual gradient method for centralized constrained optimization and the Wang-Elia method for distributed unconstrained optimization. It is shown that the optimal primal-dual point is a semiglobally exponentially stable equilibrium for the algorithm, which implies linear convergence. The analysis is based on the separation between a slow centralized optimization dynamics describing the evolution of the average estimate toward the optimum, and a fast dynamics describing the evolution of the consensus error over the network. These two dynamics are mutually coupled, and the stability analysis builds on control theoretic tools such as time-scale separation, Lyapunov theory, and the small-gain principle. Our analysis approach highlights that the consensus dynamics can be seen as a fast, parasite one, and that stability of the distributed algorithm is obtained as a robustness consequence of the semiglobal exponential stability properties of the centralized method. This perspective can be used to enable other significant extensions, such as time-varying networks or delayed communication, that can be seen as ``perturbations" of the centralized algorithm.