Stateful stream processing engines are critical for real-time analytics but lack efficient mechanisms for runtime elasticity. The dominant "stop-the-world" model, used by systems like Apache Flink, requires halting applications globally for a long time, while recent on-the-fly protocols introduce severe trade-offs: proactive approaches impose a continuous resource tax by constantly replicating state, and existing reactive solutions suffer from architectural complexity and external dependencies. This paper introduces Fugue, a novel, self-contained reactive protocol that provides seamless and resource-efficient elasticity. The core of Fugue is a two-phase design that combines a pre-emptive background state transfer with an atomic, lightweight barrier-based cutover. By moving the bulk of an operator's state off the critical path and unifying the final ownership transfer with the system's native exactly-once synchronization mechanism, Fugue guarantees correctness with minimal disruption and steady-state overhead. We implemented Fugue in Apache Flink and our evaluation on realistic benchmarks shows it reconfigures a running query up to 40x faster than native Flink while maintaining over 95% of peak throughput. Compared to state-of-the-art baselines, Fugue reduces end-to-end migration latency by up to 14x, demonstrating a superior balance of robustness, performance, and operational simplicity.