The accurate and automatic extraction of roads from satellite imagery is critical for applications in navigation and urban planning, significantly reducing the need for manual annotation. Many existing methods decompose this task into keypoint extraction and connectedness prediction, but often struggle to capture long-range dependencies and complex topologies. Here, we propose LineGraph2Road, a framework that improves connectedness prediction by formulating it as binary classification over edges in a constructed global but sparse Euclidean graph, where nodes are keypoints extracted from segmentation masks and edges connect node pairs within a predefined distance threshold, representing potential road segments. To better learn structural link representation, we transform the original graph into its corresponding line graph and apply a Graph Transformer on it for connectedness prediction. This formulation overcomes the limitations of endpoint-embedding fusion on set-isomorphic links, enabling rich link representations and effective relational reasoning over the global structure. Additionally, we introduce an overpass/underpass head to resolve multi-level crossings and a coupled NMS strategy to preserve critical connections. We evaluate LineGraph2Road on three benchmarks: City-scale, SpaceNet, and Global-scale, and show that it achieves state-of-the-art results on two key metrics, TOPO-F1 and APLS. It also captures fine visual details critical for real-world deployment. We will make our code publicly available.