We consider a mechanism design problem for the joint flow control and multipath routing in informationally decentralized networks with strategic agents. Based on a surrogate optimization approach, we propose an incentive mechanism that strongly implements the social-welfare maximizing outcome in Nash equilibria. This mechanism possesses several other desirable properties, including individual rationality and budget balance at equilibrium. More importantly, in contrast to the existing literature on the network resource allocation mechanisms, the proposed mechanism is dynamically stable, meaning that the Nash equilibrium (NE) of the game induced by the mechanism can be learned by the agents in a decentralized manner. To establish dynamic stability, we propose a decentralized iterative process that always converges to a NE of the game induced by the mechanism, provided that all strategic agents follow the process. To the best of our knowledge, this is the first incentive mechanism that simultaneously possesses all the above-mentioned properties.