People living with HIV (PLWH) on suppressive antiretroviral therapy (ART) can have residual immune dysfunction and often display poorer responses to vaccination. We assessed in a cohort of PLWH (n=110) and HIV negative controls (n=64) the humoral and spike-specific B-cell responses following 1, 2 or 3 SARS-CoV-2 vaccine doses. PLWH had significantly lower neutralizing antibody (nAb) titers than HIV-negative controls at all studied timepoints. Moreover, their neutralization breadth was reduced with fewer individuals developing a neutralizing response against the Omicron variant (BA.1) relative to controls. We also observed a delayed development of neutralization in PLWH that was underpinned by a reduced frequency of spike-specific memory B cells (MBCs) and pronounced B cell dysfunction. Improved neutralization breadth was seen after the third vaccine dose in PLWH but lower nAb responses persisted and were associated with global, but not spike-specific, MBC dysfunction. In contrast to the inferior antibody responses, SARS-CoV-2 vaccination induced robust T cell responses that cross-recognized variants in PLWH. Strikingly, a subset of PLWH with low or absent neutralization had detectable functional T cell responses. These individuals had reduced numbers of circulating T follicular helper cells and an enriched population of CXCR3+CD127+CD8+ T cells after two doses of SARS-CoV-2 vaccination, which may compensate for sub-optimal serological responses in the event of infection. Therefore, normalisation of B cell homeostasis could improve serological responses to vaccines in PLWH and evaluating T cell immunity could provide a more comprehensive immune status profile in these individuals and others with B cell imbalances.