Carotid artery stenosis is a risk factor for ischemic stroke. Surgical treatment is often used to improve cerebral perfusion and prevent the development of cerebrovascular pathology and related cognitive impairment. The aim of this prospective pilot study was to evaluate the cognitive functions (CF) of patients after surgery (open or endovascular intervention) on the internal carotid artery. The study included 90 patients (mean age 62 years, 71% of men) with atherosclerotic lesions of the carotid arteries. The CF was evaluated at four time points (before the intervention, 3, 6, and 9 months after) using cognitive scales and measuring cognitive evoked potentials. The state of the brain substance before and after the intervention was evaluated by the results of diffusion-weighted magnetic resonance imaging (DW-MRI). Three and six months after the operation, the patients demonstrated minor and varied CF alterations by the MMSE scale, but by the end of the observation period (9 months) the participants had their CF at the level close to that registered before the operation (p = 0.43). Thus, the intervention-associated changes in CF, regardless of the surgical approach, were primarily transient in nature. The rare cases of CF deterioration, as registered by the postoperative DW-MRI scans, were linked to the acute brain ischemia, both symptomatic and asymptomatic, and a perioperative stroke (1 case). Advanced age and altered cerebral arteries may be listed as the risk factors for the probable CF deterioration. Evaluation of the connections between CF alterations and multiple cases of intraoperative cerebral vascular embolism requires a longer observation period.