We address an optimal mass transportation problem by means of optimal stochastic control. We consider a stochastic control problem which is a natural extension of the Monge-Kantorovich problem. Using a vanishing viscosity argument we provide a probabilistic proof of two fundamental results in mass transportation: the Kantorovich duality and the graph property for the support of an optimal measure for the Monge-Kantorovich problem. Our key tool is a stochastic duality result involving solutions of the Hamilton-Jacobi-Bellman PDE.