Reactive collision avoidance is essential for agile robots navigating complex and dynamic environments, enabling real-time obstacle response. However, this task is inherently challenging because it requires a tight integration of perception, planning, and control, which traditional methods often handle separately, resulting in compounded errors and delays. This paper introduces a novel approach that unifies these tasks into a single reactive framework using solely onboard sensing and computing. Our method combines nonlinear model predictive control with adaptive control barrier functions, directly linking perception-driven constraints to real-time planning and control. Constraints are determined by using a neural network to refine noisy RGB-D data, enhancing depth accuracy, and selecting points with the minimum time-to-collision to prioritize the most immediate threats. To maintain a balance between safety and agility, a heuristic dynamically adjusts the optimization process, preventing overconstraints in real time. Extensive experiments with an agile quadrotor demonstrate effective collision avoidance across diverse indoor and outdoor environments, without requiring environment-specific tuning or explicit mapping.