Recently, neural control policies have outperformed existing model-based planning-and-control methods for autonomously navigating quadrotors through cluttered environments in minimum time. However, they are not perception aware, a crucial requirement in vision-based navigation due to the camera's limited field of view and the underactuated nature of a quadrotor. We propose a learning-based system that achieves perception-aware, agile flight in cluttered environments. Our method combines imitation learning with reinforcement learning (RL) by leveraging a privileged learning-by-cheating framework. Using RL, we first train a perception-aware teacher policy with full-state information to fly in minimum time through cluttered environments. Then, we use imitation learning to distill its knowledge into a vision-based student policy that only perceives the environment via a camera. Our approach tightly couples perception and control, showing a significant advantage in computation speed (10 times faster) and success rate. We demonstrate the closed-loop control performance using hardware-in-the-loop simulation.