In many years, researchers desire to evaluate system reliability accurately and efficiently. This paper presents a generalized Complementary Intersection Method (CIM) for system reliability analysis for series, parallel, and mixed systems. The CIM provides an innovative way to evaluate system reliability by decomposing the probability of a high-order joint failure event into probabilities of complementary intersection events. However, its application has been limited to a series system only. In this paper, a generalized CIM framework is proposed for system reliability analysis with any system structure (or configuration) (e.g., series, parallel, and mixed). The CIM is generalized for both parallel and mixed system reliability. The System Structure Matrix (SS-Matrix) is proposed to characterize any system structure in a comprehensive manner. The Binary Decision Diagram (BDD) technique is employed to identify system's mutually exclusive path sets, of which each path set is a series system. On consequence, system reliability with any system structure is decomposed into the probabilities of the mutually exclusive path sets, which can be evaluated using different reliability analysis methods. Three examples are used to demonstrate the uniqueness and effectiveness of the proposed methodology.