The epidemic control approach was based on non-pharmacological measures in the first year of the COVID-19 pandemic, followed by vaccine uptake in the second year. Vaccine uptake depends on the individual attitude toward vaccination. The aim was to assess the changes in attitudes regarding COVID-19 vaccine protection during the pandemic and to determine the vaccination uptake concerning these attitudes. A panel study on COVID-19 vaccine attitudes and vaccination against COVID-19 was conducted in Belgrade, Serbia. The first survey was carried out in May–June 2020, and the second survey was organized in August–September 2021. During the baseline testing performed in 2020, 64.4% of respondents believed that the future vaccine against COVID-19 could protect against the COVID-19 disease, while 9.7% thought that it could not, and 25.9% were unsure. One year later, in the second survey, the percentage of participants with positive attitudes was slightly lower (64.7% vs. 62.5%). However, negative attitudes turned positive in 34% of cases, and 28.9% became unsure about vaccine protection (p < 0.001). Out of the 390 participants included in the study, 79.7% were vaccinated against COVID-19 until follow-up. There is a statistically significant difference in vaccination uptake compared to the baseline attitude about the protection of the COVID-19 vaccine. The main finding of our study is that the majority of participants who were vaccine hesitant during the baseline testing changed their opinion during the follow-up period. Additionally, the baseline attitude about the protection of the COVID-19 vaccine has been shown to be a potential determinant of vaccination uptake.