Cognitive reasoning is thought to require functional interactions between whole-brain networks. Such networks rely on both cerebral hemispheres, with the corpus callosum providing cross-hemispheric communication. Here we used high-field functional magnetic resonance imaging (7 T fMRI), a well validated cognitive task, and brain network analyses to investigate the functional networks underlying cognitive reasoning in individuals with corpus callosum dysgenesis (CCD), an anatomical abnormality that affects the corpus callosum. Participants with CCD were asked to solve cognitive reasoning problems while their brain activity was measured using fMRI. The complexity of these problems was parametrically varied by changing the complexity of relations that needed to be established between shapes within each problem matrix. Behaviorally, participants showed a typical reduction in task performance as problem complexity increased. Task-evoked neural activity was observed in brain regions known to constitute two key cognitive control systems: the fronto-parietal and cingulo-opercular networks. Under low complexity demands, network topology and the patterns of local neural activity in the CCD group closely resembled those observed in neurotypical controls. By contrast, when asked to solve more complex problems, participants with CCD showed a reduction in neural activity and connectivity within the fronto-parietal network. These complexity-induced, as opposed to resting-state, differences in functional network activity help resolve the apparent paradox between preserved network architecture found at rest in CCD individuals, and the heterogeneous deficits they display in response to cognitive task demands [preprint: https://doi.org/10.1101/312629].