Optimal heuristic search has been successful in many domains where the cost of each action can easily be obtained. However, in many problems, the exact edge cost is expensive to compute. Existing search algorithms face a significant performance bottleneck, due to an excessive overhead associated with dynamically calculating exact edge costs. We present DEA∗, an algorithm for problems with expensive edge cost computations. DEA∗ combines heuristic edge cost evaluations with delayed node expansions, reducing the number of exact edge computations. We formally prove that DEA∗ is optimal and it is efficient with respect to the number of exact edge cost computations. We empirically evaluate DEA∗ on multiple-worker routing problems where the exact edge cost is calculated by invoking an external multi-modal journey planning engine. The results demonstrate the effectiveness of our ideas in reducing the computational time and improving the solving ability. In addition, we show the advantages of DEA∗ in domain-independent planning, where we simulate that accurate edge costs are expensive to compute.