Various networks such as cloud computing, water/gas/electricity networks, wireless sensor networks, transportation networks, and 4G/5G networks, have become an integral part of our daily lives. A binary-state network (BN) is often used to model network structures and applications. The BN reliability is the probability that a BN functions continuously; that is, that there is always a simple path connected between a specific pair of nodes. This metric is a popular index for designing, managing, controlling, and evaluating networks. The traditional BN reliability assumes that the reliability of each arc is known in advance. However, this is not always the case. Functioning components operate under different environments; moreover, a network might have newly installed components. Hence, the reliability of these components is not always known. To resolve the aforementioned problems, in which the reliability of some components of a network are uncertain, we introduce the fuzzy concept for the analysis of these components, and propose a new algorithm to solve this uncertainty-component BN reliability problem. The time complexity of the proposed algorithm is analyzed, and the superior performance of the algorithm is demonstrated through examples.