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 messenger RNA–vaccinated health care workers (HCWs). Twenty-three HCWs 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 HCWs 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 were consistently two- to fourfold lower than those against 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. 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.