This paper presents a new solution to the Dynamic All-Pairs Shortest Path Routing Problem, using a pursuit estimator learning approach. It involves finding the shortest path in a stochastic network, where there are continuous probabilistically-based updates in link-costs. In this paper we present the details of the algorithm and also provide an example to illustrate how the algorithm would function. The experimental results of the algorithm show that the algorithm is few orders of magnitude superior to the algorithms available in the literature. It can be used to find the shortest path (between all pairs of nodes in a network) within the "statistical" average network, which converges irrespective of whether there are new changes in link-costs or not. On the other hand, the existing algorithms will fail to exhibit such a behavior and would recalculate the affected shortest paths after each link-cost update.