Vaccine breakthrough SARS-CoV-2 infections necessitating hospitalization have emerged as a relevant problem with longer time interval since vaccination and the predominance of the Delta variant. The aim of this study was to evaluate the association between primary vaccination with four SARS-CoV-2 vaccines authorized for use in the European Union—BNT162b2, ChAdOx-1S, mRNA-1273 or Ad.26.COV2.S—and progression to critically severe disease (mechanical ventilation or death) and duration of hospitalization among adult patients with PCR-confirmed acute COVID-19 hospitalized during the Delta variant predominance (October–November 2021) in Slovenia. Among the 529 enrolled patients hospitalized with COVID-19 (median age, 65 years; 58.2% men), 175 (33.1%) were fully vaccinated at the time of symptom onset. Compared with 345 unvaccinated patients, fully vaccinated patients with breakthrough infections were older, more often immunocompromised, and had higher Charlson comorbidity index scores. After adjusting for sex, age, and comorbidities, fully vaccinated patients had lower odds for progressing to critically severe disease and were discharged from the hospital earlier than unvaccinated patients. Vaccination against SARS-CoV-2 remains an extremely effective intervention to alleviate morbidity and mortality in COVID-19 patients.