Terminal-pair Reliability (TR) in network management determines the probabilistic reliability between two nodes (the source and sink) of a network, given failure probabilities of all links. It has been shown that TR can be effectively computed by means of the network reduction technique. Existing reduction axioms, unfortunately, are limited to simple rules such as valueless link removal and series-parallel link reduction. In this paper, we propose a novel reduction axiom, referred to as triangle reduction. The triangle reduction axiom transforms a graph containing a triangle subgraph to that excluding the base of the triangle. The computational complexity of the transformation is as low as O(1). The paper further provides an assessment of the effectiveness of triangle reduction on partition-based TR algorithms with respect to the number of subproblems and computation time. Experimental results demonstrate that, incorporating triangle reduction, the partition-based TR algorithms yield a substantially reduced number of subproblems and computation time for all benchmarks and random networks.