Dynamic obstacle avoidance (DOA) is critical for quadrupedal robots operating in environments with moving obstacles or humans. Existing approaches typically rely on navigation-based trajectory replanning, which assumes sufficient reaction time and leading to fails when obstacles approach rapidly. In such scenarios, quadrupedal robots require reflexive evasion capabilities to perform instantaneous, low-latency maneuvers. This paper introduces Reflexive Evasion Robot (REBot), a control framework that enables quadrupedal robots to achieve real-time reflexive obstacle avoidance. REBot integrates an avoidance policy and a recovery policy within a finite-state machine. With carefully designed learning curricula and by incorporating regularization and adaptive rewards, REBot achieves robust evasion and rapid stabilization in instantaneous DOA tasks. We validate REBot through extensive simulations and real-world experiments, demonstrating notable improvements in avoidance success rates, energy efficiency, and robustness to fast-moving obstacles. Videos and appendix are available on https://rebot-2025.github.io/.