Aerial insects exhibit highly agile maneuvers such as sharp braking, saccades, and body flips under disturbance. In contrast, insect-scale aerial robots are limited to tracking non-aggressive trajectories with small body acceleration. This performance gap is contributed by a combination of low robot inertia, fast dynamics, uncertainty in flapping-wing aerodynamics, and high susceptibility to environmental disturbance. Executing highly dynamic maneuvers requires the generation of aggressive flight trajectories that push against the hardware limit and a high-rate feedback controller that accounts for model and environmental uncertainty. Here, through designing a deep-learned robust tube model predictive controller, we showcase insect-like flight agility and robustness in a 750-millgram flapping-wing robot. Our model predictive controller can track aggressive flight trajectories under disturbance. To achieve a high feedback rate in a compute-constrained real-time system, we design imitation learning methods to train a two-layer, fully connected neural network, which resembles insect flight control architecture consisting of central nervous system and motor neurons. Our robot demonstrates insect-like saccade movements with lateral speed and acceleration of 197 centimeters per second and 11.7 meters per second square, representing 447$\%$ and 255$\%$ improvement over prior results. The robot can also perform saccade maneuvers under 160 centimeters per second wind disturbance and large command-to-force mapping errors. Furthermore, it performs 10 consecutive body flips in 11 seconds - the most challenging maneuver among sub-gram flyers. These results represent a milestone in achieving insect-scale flight agility and inspire future investigations on sensing and compute autonomy.