Numerous neuroimaging studies have identified various brain networks using task-free analyses. While these networks undoubtedly support higher cognition, their precise functional characteristics are rarely probed directly. The frontal, temporal, and parietal lobes contain the majority of the tertiary association cortex, which are key substrates for higher cognition including executive function, language, memory, and attention. Accordingly, we established the cognitive signature of a set of contrastive brain networks on the main tertiary association cortices, identified in two task-independent datasets. Using graph-theory analysis, we revealed multiple networks across the frontal, temporal, and parietal cortex, derived from structural and functional connectivity. The patterns of network activity were then investigated using three task-active fMRI datasets to generate the functional profiles of the identified networks. We employed representational dissimilarity analysis on these functional data to quantify and compare the representational characteristics of the networks. Our results demonstrated that the topology of the task-independent networks was strongly associated with the patterns of network activity in the task-active fMRI. Our findings establish a direct relationship between the brain networks identified from task-free datasets and higher cognitive functions including cognitive control, language, memory, visuospatial function, and perception. Not only does this study support the widely held view that higher cognitive functions are supported by widespread, distributed cortical networks, but also it elucidates a methodological approach for formally establishing their relationship.