Aging is associated with gradual changes in cognition, yet some individuals exhibit protection against aging-related cognitive decline. The topological characteristics of brain networks that support protection against cognitive decline in aging are unknown. Here, we investigated whether the robustness of brain networks, queried via the delineation of the brain's core network structure, supports superior cognitive performance in healthy aging individuals (n=320, ages 60-90). First, we decomposed each subject's functional brain networks using k-shell decomposition, finding that cognitive function is associated with more robust connectivity of core nodes, primarily within the frontoparietal control network (FPCN). Next, we find that the resilience of core brain network nodes, within the FPCN in particular, relates to cognition. Finally, we show that the degree of segregation in functional networks mediates relationships between network resilience and cognition. Together, these findings suggest that brain networks balance between robust core connectivity and segregation to facilitate high cognitive performance in aging.