Chronic exposure to seizures in patients with left hemisphere (LH) epileptic focus could favor higher activation in the contralateral hemisphere during language processing, but the cognitive effects of this remain unclear. This study assesses the relationship between asymmetry in hemispheric activation during language fMRI and performance in verbal and non-verbal tasks. Whereas prior studies primarily used fMRI paradigms that favor frontal lobe activation and less prominent activation of the medial or superior temporal lobes, we used a verbal comprehension paradigm previously demonstrated to activate reliably receptive language areas. Forty-seven patients with drug-resistant epilepsy candidates for surgery underwent a multidisciplinary assessment, including a comprehensive neuropsychological evaluation and an fMRI verbal comprehension paradigm. Patients were distributed in two groups depending on laterality indexes (LI): typical hemispheric asymmetry (unilateral left activation preponderance; n = 23) and atypical hemispheric asymmetry (bilateral or unilateral right preponderance; n = 24). Right-handedness and right hemisphere (RH) focus were significant predictors of typical asymmetry. Patients with typical activation pattern presented better performance intelligence quotient and verbal learning than patients with atypical hemispheric asymmetry (for all, p < 0.014). Patients with LH focus had more frequently atypical hemispheric asymmetry than patients with RH focus (p = 0.05). Specifically, they showed lower LI and this was related to worse performance in verbal and non-verbal tasks. In conclusion, an increased activation of homologous RH areas for verbal comprehension processing could imply a competition of cognitive resources in the performance of the same task, disrupting cognitive performance.