Rotor failures in quadrotors may result in high-speed rotation and vibration due to rotor imbalance, which introduces significant challenges for autonomous flight in unknown environments. The mainstream approaches against rotor failures rely on fault-tolerant control (FTC) and predefined trajectory tracking. To the best of our knowledge, online failure detection and diagnosis (FDD), trajectory planning, and FTC of the post-failure quadrotors in unknown and complex environments have not yet been achieved. This paper presents a rotor-failure-aware quadrotor navigation system designed to mitigate the impacts of rotor imbalance. First, a composite FDD-based nonlinear model predictive controller (NMPC), incorporating motor dynamics, is designed to ensure fast failure detection and flight stability. Second, a rotor-failure-aware planner is designed to leverage FDD results and spatial-temporal joint optimization, while a LiDAR-based quadrotor platform with four anti-torque plates is designed to enable reliable perception under high-speed rotation. Lastly, extensive benchmarks against state-of-the-art methods highlight the superior performance of the proposed approach in addressing rotor failures, including propeller unloading and motor stoppage. The experimental results demonstrate, for the first time, that our approach enables autonomous quadrotor flight with rotor failures in challenging environments, including cluttered rooms and unknown forests.