In everyday life, information from different cognitive domains—such as visuospatial attention, alertness and inhibition—needs to be integrated between different brain regions. Early models suggested that completely segregated brain networks control these three cognitive domains. However, more recent accounts, mainly based on neuroimaging data in healthy participants, indicate that different tasks lead to specific patterns of activation within the same, higher-order and ‘multiple-demand’ network. If so, then a lesion to critical substrates of this common network should determine a concomitant impairment in all three cognitive domains. The aim of the present study was to critically investigate this hypothesis, i.e. to identify focal stroke lesions within the network that can concomitantly affect visuospatial attention, alertness and inhibition. We studied an unselected sample of 60 first-ever right-hemispheric, subacute stroke patients using a data-driven, bottom-up approach. Patients performed 12 standardized neuropsychological and oculomotor tests, four per cognitive domain. A principal component analysis revealed a strong relationship between all three cognitive domains: 10 of 12 tests loaded on a first, common component. Analysis of the neuroanatomical lesion correlates using different approaches (i.e. voxel-based and tractwise lesion-symptom mapping, disconnectome maps) provided convergent evidence on the association between severe impairment of this common component and lesions at the intersection of superior longitudinal fasciculus II and III, frontal aslant tract and, to a lesser extent, the putamen and inferior fronto-occipital fasciculus. Moreover, patients with a lesion involving this region were significantly more impaired in daily living cognition, which provides an ecological validation of our results. A probabilistic functional atlas of the multiple-demand network was performed to confirm the potential relationship between patients’ lesion substrates and observed cognitive impairments as a function of the multiple-demand network connectivity disruption. These findings show, for the first time, that a lesion to a specific white matter crossroad can determine a concurrent breakdown in all three considered cognitive domains. Our results support the multiple-demand network model, proposing that different cognitive operations depend on specific collaborators and their interaction, within the same underlying neural network. Our findings also extend this hypothesis by showing (i) the contribution of superior longitudinal fasciculus and frontal aslant tract to the multiple-demand network; and (ii) a critical neuroanatomical intersection, crossed by a vast amount of long-range white matter tracts, many of which interconnect cortical areas of the multiple-demand network. The vulnerability of this crossroad to stroke has specific cognitive and clinical consequences; this has the potential to influence future rehabilitative approaches.