The emergence of SARS-CoV-2 variants harboring mutations in the spike (S) protein has raised concern about potential immune escape. Here, we studied humoral and cellular immune responses to wild type SARS-CoV-2 and the B.1.1.7 and B.1.351 variants of concern in a cohort of 121 BNT162b2 mRNA-vaccinated health care workers (HCW). Twenty-three HCW recovered from mild COVID-19 disease and exhibited a recall response with high levels of SARS-CoV-2-specific functional antibodies and virus-specific T cells after a single vaccination. Specific immune responses were also detected in seronegative HCW after one vaccination, but a second dose was required to reach high levels of functional antibodies and cellular immune responses in all individuals. Vaccination-induced antibodies cross-neutralized the variants B.1.1.7 and B.1.351, but the neutralizing capacity and Fc-mediated functionality against B.1.351 was consistently 2- to 4-fold lower than to the homologous virus. In addition, peripheral blood mononuclear cells were stimulated with peptide pools spanning the mutated S regions of B.1.1.7 and B.1.351 to detect cross-reactivity of SARS-CoV-2-specific T cells with variants. Importantly, we observed no differences in CD4+ T-cell activation in response to variant antigens, indicating that the B.1.1.7 and B.1.351 S proteins do not escape T-cell-mediated immunity elicited by the wild type S protein. In conclusion, this study shows that some variants can partially escape humoral immunity induced by SARS-CoV-2 infection or BNT162b2 vaccination, but S-specific CD4+ T-cell activation is not affected by the mutations in the B.1.1.7 and B.1.351 variants.