The coronavirus disease 2019 (COVID-19) can disrupt various brain functions. Over a one-year period, we aimed to assess brain activity and cognitive function in 53 COVID-19 patients and 30 individuals without COVID-19 (or asymptomatic). The Montreal Cognitive Assessment, Trail Making Test Parts A and B (TMT-A and B), and Digit Span Test were used to assess cognitive function. Cognitive variables and electroencephalography (EEG) data (activity, mobility, and complexity) were compared between the groups at rest and during cognitive demand (F3-F7, Fz-F3, Fz-F4, and F4-F8). There was a reduction in F3-F7 activity during the TMT-B in the COVID-19 group at 6-12 months compared to the controls (p = 0.01) at baseline (p = 0.03), a reduction in signal complexity at F3-F7 at rest in the COVID-19 group at baseline and 6-12 months compared to the controls (p < 0.001), and a reduction in Fz-F4 activity at rest from 6-12 months in the post-COVID group compared to baseline (p = 0.02) and 3-6 months (p = 0.04). At 6-12 months, there was a time increase in TMT-A in the COVID-19 group compared to that in the controls (p = 0.04). Some correlations were found between EEG data and cognitive test in both groups. In conclusion, there was a reduction in brain activity at rest in the Fz-F4 areas and during high cognitive demands in the F3-F7 areas. A reduction in signal complexity in F3-F7 at rest was found in the COVID-19 group at 6-12 months after acute infection. Furthermore, individuals with COVID-19 experience long-term changes in cognitive function.