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 = <0.0001), and between those treated and untreated with GA, no statistically significant differences were found. Regarding oxidation, no statistically significant differences were detected. Upon categorizing the pwMS into cognitively impaired and cognitively preserved groups, the lactate was elevated in the pwMS with cognitive preservation (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.