Narrative discourse (ND) comprehension is a complex task that implies not only linguistic abilities but also other cognitive abilities, including efficient executive functioning. An executive dysfunction has been described in dementia with Lewy bodies (DLB) from the early stage. Here, we question the link between executive dysfunction in DLB and narrative comprehension. The aim of our study was to evaluate ND comprehension and to investigate the neuroanatomical basis for its impairment in the early stage of DLB. DLB patients (N = 26) and controls (N = 19) underwent the ND comprehension test of the Montreal Protocol for Evaluation of Communication (MEC). An additional, qualitative analysis was conducted on their verbal productions. Cognitive tests assessing verbal episodic memory, executive functions, naming and oral syntactic comprehension were also performed. Brain gray matter correlates of the ND comprehension test were examined using voxel-based morphometry (VBM). An ND comprehension impairment was found for prodromal and mild DLB patients as compared to controls. These difficulties were correlated with the Frontal Assessment Battery (FAB) score. ND comprehension impairment in DLB was further characterized by a deficit in the organization and the logic of the discourse. Moreover, VBM analysis revealed a correlation between striatal gray matter volumes and DLB patients’ ability to extract and organize relevant information (p < 0.05, FDR correction, cluster level). The ND comprehension impairment in DLB patients could be related to their executive dysfunction through a deficit of information selection and organization that correlates with the volumetric reduction of striatal gray matter.