Real-time damping estimation for a dominant inter-area mode is important for situational awareness of potential angular instability in power systems. Electromechanical oscillations energized by large disturbances often manifest obvious nonlinearities in measurements on first several swings. Traditional methods based on linear system theory often discard first several swings intentionally to avoid nonlinearity; if not, the estimated damping ratios often vary with the length and starting point of the measuring window. By identifying a nonlinear oscillator to fit a dominant mode, this paper proposes a new measurement-based approach utilizing complete post-disturbance data for robust damping estimation independent of the measuring window. Case studies on the IEEE 9-bus system and a 48-machine Northeast Power Coordinating Council system validate the proposed approach for providing accurate and robust damping estimation compared with existing methods including the Prony's method. Meanwhile, three factors influencing damping estimation in practical applications are also addressed, including measurement noises, limited coverage of PMU measurements, and existence of multiple dominant modes.