The development of the computing power network has brought about a revolutionary effect on network routing architecture. As a result, the computing-aware network routing problem has been raised to explore routing various computational tasks to appropriate computing resources in the dynamic network. In this study, we propose a heuristic-based computing-aware routing algorithm to achieve the optimal routing path by considering the dynamic network performance and computing resource status simultaneously. Our proposed approach models the dynamic network using time-varying node and edge weights, which are obtained by mapping basic performance indicators to weights according to quality-of-service requirements. This allows us to improve the user’s experience more effectively during the routing process. Moreover, a novel heuristic-based algorithm, which creatively transforms the computing-aware routing problem into a single-source shortest path problem, has been designed to achieve the comprehensive optimal routing path. The experimental results, based on both simulated networks and a real dedicated network in Zhejiang, demonstrate that our proposed method can obtain the comprehensive optimal routing path with a lower computing time cost than enumerating search. Furthermore, our proposed computing-aware routing method has been proven to be robust to the dynamics of the network, computing resources, and service load changes.