This paper deals with the constrained DAG shortest path problem (CDSP), which finds the shortest path on a given directed acyclic graph (DAG) under any logical constraints posed on taken edges. There exists a previous work that uses binary decision diagrams (BDDs) to represent the logical constraints, and traverses the input DAG and the BDD simultaneously. The time and space complexity of this BDD-based method is derived from BDD size, and tends to be fast only when BDDs are small. However, since it does not prioritize the search order, there is considerable room for improvement, particularly for large BDDs. We combine the well-known A* search with the BDD-based method synergistically, and implement several novel heuristic functions. The key insight here is that the ‘shortest path’ in the BDD is a solution of a relaxed problem, just as the shortest path in the DAG is. Experiments, particularly practical machine learning applications, show that the proposed method decreases search time by up to 2 orders of magnitude, with the specific result that it is 2,000 times faster than a commercial solver. Moreover, the proposed method can reduce the peak memory usage up to 40 times less than the conventional method.