This paper proposes a unique and robust UAV autonomous navigation approach along one side of overhead transmission lines for inspection. To this end, we establish a perspective model and develop a novel Pan/Tilt monocular-based navigation scheme. Simultaneously, the following three key issues are addressed. First, to locate the effective landmark – transmission tower timely and reliably, we customize a neural network for tower detection and combine it with a fast and smooth tracking. Second, to provide UAV with a robust and precise heading, we detect the transmission lines and compute and optimize their vanishing point. Third, to keep a safe distance from transmission lines, we optimize a homography matrix to restore the parallel nature of transmission lines and perceive the distance variation by a point set registration model. Finally, by the designed UAV platform, we test the whole system in a real-world transmission-line inspection scenario under different weather condition and achieve an encouraging result. Our approach provides great flexibility for refined inspection and effectively improves inspection safety.