We present the calculation of next-to-next-to-leading order (NNLO) corrections in perturbative QCD for the production of a Higgs boson decaying into a pair of bottom quarks in association with a leptonically decaying weak vector boson: pp→VH+X→ℓℓ¯bb¯¯¯+X. We consider the corrections to both the production and decay sub-processes, retaining a fully differential description of the final state including off-shell propagators of the Higgs and vector boson. The calculation is carried out using the antenna subtraction formalism and is implemented in the NNLOjet framework. Clustering and identification of b-jets is performed with the flavour-kt algorithm and results for fiducial cross sections and distributions are presented for the LHC at s√ = 13 TeV. We assess the residual theory uncertainty by varying the production and decay scales independently and provide scale uncertainty bands in our results, yielding percent-level accurate predictions for observables in this Higgs production mode computed at NNLO. Confronting a na¨ıve perturbative expansion of the cross section against the customary re-scaling procedure to a fixed branching ratio reveals that starting from NNLO, the latter could be inadequate in estimating missing higher-order effects through scale variations.