Rapid prediction of Road Network Functionality (RNF) during extreme rainfall-induced flooding is crucial for supporting proactive and real-time emergency planning, such as rescue, evacuation planning, and emergency supply distribution. Unlike normal operational conditions, extreme rainfall events introduce complex non-stationary, non-Euclidean characteristics to RNF due to intricate meteorological and hydrological processes, as well as the role of a community's road network in emergency response planning. Conventional physics-based flood simulations and flow-based road network analyses typically lack the computational efficiency required for real-time RNF predictions, hindering timely risk mitigation decisions. This study leverages the accuracy of physics-based simulations and the efficacy of deep-learning technologies to develop a deep learning-based surrogate model for Rain-to-RNF (R2R) predictions. This model couples Long Short-Term Memory (LSTM) networks with Spatial-Temporal Graph Convolutional Networks (ST-GCNs) to uniquely capture the spatiotemporal dynamics of RNF under extreme rainfall events. The predictive accuracy, stability, and versatility of the R2R surrogate model are demonstrated in four flood-prone communities in Zhejiang Province. Its implementation during Typhoon Fitow (2013) over a 30-hour intense rainfall showcases its promising predictive capacity and unparalleled computational efficiency. This research advances disaster management, enhancing the resilience and responsiveness of community infrastructure during extreme weather events.