The long-term effects of coronavirus disease 2019 (COVID-19) infection are not fully known. In this study, we aimed to evaluate cognitive function and mood changes with 1-year follow-up in the elderly after COVID-19 disease. Ninety COVID-19 survivors and 90 healthy controls were included in the study between April 2022 and 2023. The patients were evaluated at the 1st, 6th, and 12th months for cognition, depression, and sleep quality. Cognitive function is assessed by the Montreal Cognitive Assessment (MoCA), sleep quality by the Pittsburgh Sleep Quality Index, and depression by the Yesavage Geriatric Depression Scale. COVID-19 survivors secured lower scores in certain domains of the MoCA in comparison with the controls at the first and sixth months. However, at the 12th month, no difference was observed in total MoCA (p = 0.100), Yesavage Geriatric Depression Scale (p = 0.503), and Pittsburgh Sleep Quality Index (p = 0.907) between survivors and controls. Older patients who recovered from COVID-19 have lower cognitive function compared with controls up to 12 months. However, cognitive function scores were similar at the end of the first year except for memory scores.