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Evaluating two‐terminal network reliability is a classical problem with numerous applications. Because this problem is #P‐Complete, practical studies involving large systems commonly resort to approximating or estimating system reliability rather than evaluating it exactly. Researchers have characterized signatures, such as the destruction spectrum and survival signature, which summarize the system's structure and give rise to procedures for evaluating or approximating network reliability. These procedures are advantageous if the signature can be computed efficiently; however, computing the signature is challenging for complex systems. With this motivation, we consider the use of Monte Carlo (MC) simulation to estimate the survival signature of a two‐terminal network in which there are two classes of i.i.d. components. In this setting, we prove that each MC replication to estimate the signature of a multi‐class system entails solving a multi‐objective maximum capacity path problem. For the case of two classes of components, we adapt a Dijkstra's‐like bi‐objective shortest path algorithm from the literature for the purpose of solving the resulting bi‐objective maximum capacity path problem. We perform computational experiments to compare our method's efficiency against intuitive benchmark approaches. Our computational results demonstrate that the bi‐objective optimization approach consistently outperforms the benchmark approaches, thereby enabling a larger number of MC replications and improved accuracy of the reliability estimation. Furthermore, the efficiency gains versus benchmark approaches appear to become more significant as the network increases in size.