Achieving both high accuracy and topological continuity in road segmentation from satellite imagery is a critical goal for applications ranging from urban planning to disaster response. State-of-the-art methods often rely on Vision Transformers, which excel at capturing global context, yet their quadratic complexity is a significant barrier to efficient deployment, particularly for on-board processing in resource-constrained platforms. In contrast, emerging State Space Models like Mamba offer linear-time efficiency and are inherently suited to modeling long, continuous structures. We posit that these architectures have complementary strengths. To this end, we introduce PathMamba, a novel hybrid architecture that integrates Mamba's sequential modeling with the Transformer's global reasoning. Our design strategically uses Mamba blocks to trace the continuous nature of road networks, preserving topological structure, while integrating Transformer blocks to refine features with global context. This approach yields topologically superior segmentation maps without the prohibitive scaling costs of pure attention-based models. Our experiments on the DeepGlobe Road Extraction and Massachusetts Roads datasets demonstrate that PathMamba sets a new state-of-the-art. Notably, it significantly improves topological continuity, as measured by the APLS metric, setting a new benchmark while remaining computationally competitive.