The critical infrastructure system exhibits a high degree of instability because of the frequent occurrence of extreme natural disasters. This study proposes a method for exploring the cascading failures of critical infrastructure systems under such impacts, based on dependency relationships for importance analysis and probability assessment. With this method, a cascading failure scenario graph of critical infrastructure is established and empirical research is conducted using typhoon rainfall events as a case study. The main contributions of this study are summarized as follows: it proposes a method for the causal identification of the interrelationships among critical infrastructures; it sorts and evaluates the importance and cascading failure probability of critical infrastructure by using the Interpretative Structural Modeling (ISM) and Rank-order Centroid method (ROC); and it generates scenario graphs of critical infrastructure cascading failures to visualize event states, environmental characteristics, and evolutionary paths. This study provides a new method and tool for disaster risk management of complex systems involved in critical infrastructure.