Successful performance of challenging cognitive tasks depends on a consistent functional segregation of activity within the default-mode network, on the one hand, and control networks encompassing frontoparietal and cingulo-opercular areas on the other. Recent work, however, has suggested that in some cognitive control contexts nodes within the default-mode and control networks may actually cooperate to achieve optimal task performance. Here, we used functional magnetic resonance imaging to examine whether the ability to relate variables while solving a cognitive reasoning problem involves transient increases in connectivity between default-mode and control regions. Participants performed a modified version of the classic Wason selection task, in which the number of variables to be related is systematically varied across trials. As expected, areas within the default-mode network showed a parametric deactivation with increases in relational complexity, compared with neural activity in null trials. Critically, some of these areas also showed enhanced connectivity with task-positive control regions. Specifically, task-based connectivity between the striatum and the angular gyri, and between the thalamus and right temporal pole, increased as a function of relational complexity. These findings challenge the notion that functional segregation between regions within default-mode and control networks invariably support cognitive task performance, and reveal previously unknown roles for the striatum and thalamus in managing network dynamics during cognitive reasoning.