This paper addresses the problem of Virtualized Network Functions placement and traffic steering in Cloud infrastructures. We design an efficient dynamic programming (DP) algorithm for joint VNF placement and traffic steering that runs in polynomial time. In compliance with dynamic programming approaches, we organize the problem in smaller interdependent subproblems (one per VNF in the requested chain) that are solved successively and aggregated to compose the overall chain placement. We demonstrate the potential of our solution by comparing it with a Multi-Stage approach and a greedy algorithm. Simulation results show that our dynamic programming algorithm achieves better performance in terms of resource usage, acceptance rate and cloud provider revenue