Australia's iconic Great Barrier Reef (GBR) continues to suffer from repeated impacts of cyclones, coral bleaching, and outbreaks of the coral-eating crown-of-thorns starfish (COTS), losing much of its coral cover in the process. This raises the question of the ecosystem's systemic resilience and its ability to rebound after large-scale population loss. Here, we reveal that around 100 reefs of the GBR, or around 3%, have the ideal properties to facilitate recovery of disturbed areas, thereby imparting a level of systemic resilience and aiding its continued recovery. These reefs (1) are highly connected by ocean currents to the wider reef network, (2) have a relatively low risk of exposure to disturbances so that they are likely to provide replenishment when other reefs are depleted, and (3) have an ability to promote recovery of desirable species but are unlikely to either experience or spread COTS outbreaks. The great replenishment potential of these 'robust source reefs', which may supply 47% of the ecosystem in a single dispersal event, emerges from the interaction between oceanographic conditions and geographic location, a process that is likely to be repeated in other reef systems. Such natural resilience of reef systems will become increasingly important as the frequency of disturbances accelerates under climate change.