(1) Background: Acute COVID-19 infections produce alterations in the skeletal muscle, leading to acute sarcopenia, but the medium- and long-term consequences are still unknown. The aim of this study was to evaluate: (1) body composition; (2) muscle strength and the prevalence of sarcopenia; and (3) the relationship between muscle strength with symptomatic and functional evolution in older patients affected by/recovered from COVID-19; (2) Methods: A prospective, longitudinal study of patients aged ≥65 years who had suffered from COVID-19 infection between 1 March and 31 May 2020, as confirmed by PCR or subsequent seroconversion. Persistent symptoms, as well as anthropometric, clinical, and analytical characteristics, were analyzed at 3 and 12 months after infection. The degree of sarcopenia was determined by dynamometry and with SARC-F; (3) Results: 106 participants, aged 76.8 ± 7 years, were included. At 3 months postinfection, a high percentage of sarcopenic patients was found, especially among women and in those with hospitalization. At 12 months postinfection, this percentage had decreased, coinciding with a functional and symptomatic recovery, and the normalization of inflammatory parameters, especially interleukin-6 (4.7 ± 11.6 pg/mL vs. 1.5 ± 2.4 pg/mL, p < 0.05). The improvement in muscle strength was accompanied by significant weight gain (71.9 ± 12.1 kg vs. 74.7 ± 12.7 kg, p < 0.001), but not by an increase in lean mass (49.6 ± 10 vs. 49.9 ± 10, p 0.29); (4) Conclusions: Older COVID-19 survivors presented a functional, clinical, and muscular recovery 12 months postinfection. Even so, it is necessary to carry out comprehensive follow-ups and assessments that include aspects of nutrition and physical activity.