Complex human cognition arises from the integrated processing of multiple brain systems. However, little is known about how brain systems and their interactions might relate to, or perhaps even explain, human cognitive capacities. Here, we address this gap in knowledge by proposing a mechanistic framework linking frontoparietal system activity, default mode system activity, and the interactions between them, with individual differences in working memory capacity. We show that working memory performance depends on the strength of functional interactions between the frontoparietal and default mode systems. We find that this strength is modulated by the activation of two newly described brain regions, and demonstrate that the functional role of these systems is underpinned by structural white matter. Broadly, our study presents a holistic account of how regional activity, functional connections, and structural linkages together support integrative processing across brain systems in order for the brain to execute a complex cognitive process.