People living with human immunodeficiency virus (PLWH) often have neurocognitive impairment. However, findings on HIV-related differences in brain network function underlying these impairments are inconsistent. One principle frequently absent from these reports is that brain function is largely emergent from brain structure. PLWH commonly have degraded white matter; we hypothesized that functional communities connected by degraded white matter tracts would show abnormal functional connectivity. We measured white matter integrity in 69 PLWH and 67 controls using fractional anisotropy (FA) in 24 intracerebral white matter tracts. Then, among tracts with degraded FA, we identified gray matter regions connected to these tracts and measured their functional connectivity during rest. Finally, we identified cognitive impairment related to these structural and functional connectivity systems. We found HIV-related decreased FA in the corpus callosum body (CCb), which coordinates activity between the left and right hemispheres, and corresponding increases in functional connectivity. Finally, we found that individuals with impaired cognitive functioning have lower CCb FA and higher CCb functional connectivity. This result clarifies the functional relevance of the corpus callosum in HIV and provides a framework in which abnormal brain function can be understood in the context of abnormal brain structure, which may both contribute to cognitive impairment.