Few studies have comprehensively compared severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) vaccine-induced and hybrid B- and T-cell responses in people with HIV (PWH) to those in comparable controls without HIV. We included 195 PWH and 246 comparable controls from the AGEhIV COVID-19 substudy. A positive nucleocapsid antibody (INgezim IgA/IgM/IgG) or self-reported PCR test defined prior SARS-CoV-2 infection. SARS-CoV-2 anti-spike (anti-S) IgG titers and anti-S IgG production by memory B cells were assessed. Neutralizing antibody titers were determined in a subset of participants. T-cell responses were assessed by gamma interferon (IFN-γ) release and activation-induced marker assay. We estimated mean differences in postvaccination immune responses (β) between levels of determinants. Anti-S IgG titers and anti-S IgG production by memory B cells were not different between PWH and controls. Prior SARS-CoV-2 infection (β = 0.77), receiving mRNA vaccine (β = 0.56), female sex (β = 0.24), fewer days between last vaccination and sampling (β = 0.07), and a CD4/CD8 ratio of <1.0 (β = -0.39) were independently associated with anti-S IgG titers, but HIV status was not. Neutralization titers against the ancestral and Delta and Omicron SARS-CoV-2 variants were not different between PWH and controls. IFN-γ release was higher in PWH. Prior SARS-CoV-2 infection (β = 2.39), HIV-positive status (β = 1.61), and fewer days between last vaccination and sampling (β = 0.23) were independently associated with higher IFN-γ release. The percentages of SARS-CoV-2-reactive CD4+ and CD8+ T cells, however, were not different between PWH and controls. Individuals with well-controlled HIV generally mount robust vaccine-induced as well as hybrid B- and T-cell immunity across SARS-CoV-2 vaccine platforms similar to controls. Determinants of a reduced vaccine response were likewise largely similar in both groups and included a lower CD4/CD8 ratio. IMPORTANCE Some studies have suggested that people with HIV may respond less well to vaccines against SARS-CoV-2. We comprehensively compared B- and T-cell responses to different COVID-19 vaccines in middle-aged persons with well-treated HIV and individuals of the same age without HIV, who were also highly comparable in terms of demographics and lifestyle, including those with prior SARS-CoV-2 infection. Individuals with HIV generally mounted equally robust immunity to the different vaccines. Even stronger immunity was observed in both groups after prior SARS-CoV-2 infection. These findings are reassuring with respect to the efficacy of SARS-Cov-2 vaccines for the sizable and increasing global population of people with HIV with access and a good response to HIV treatment.