Multiple sclerosis (MS) is an autoimmune disease characterized by demyelination and neuroinflammation, often accompanied by cognitive impairment. This study aims (1) to investigate the potential of glatiramer acetate (GA) as a therapy for preventing cognitive decline in patients with MS (pwMS) by modulating oxidative stress (OS) and (2) to seek out the differences in cognition between pwMS in a cohort exhibiting good clinical evolution and control subjects (CS). An exploratory, prospective, multicentre, cross-sectional case–control study was conducted, involving three groups at a 1:1:1 ratio—41 GA-treated pwMS, 42 untreated pwMS, and 42 CS. The participants performed a neuropsychological battery and underwent venepuncture for blood sampling. The inclusion criteria required an Expanded Disability Status Scale score of ≤3.0 and a minimum of 5 years of MS disease. Concerning cognition, the CS had a better performance than the pwMS (p = p = 0.038). The pwMS exhibited a worse cognitive performance than the CS. The pwMS treated with GA did not show an improvement in oxidation. Lactate emerged as a potential biomarker for cognitive preservation.