A search for the decay of the Standard Model Higgs boson into a bb¯ pair when produced in association with a W or Z boson is performed with the ATLAS detector. The analysed data, corresponding to an integrated luminosity of 36.1 fb−1, were collected in proton-proton collisions in Run 2 of the Large Hadron Collider at a centre-of-mass energy of 13 TeV. Final states containing zero, one and two charged leptons (electrons or muons) are considered, targeting the decays Z → νν, W → ℓν and Z → ℓℓ. For a Higgs boson mass of 125 GeV, an excess of events over the expected background from other Standard Model processes is found with an observed significance of 3.5 standard deviations, compared to an expectation of 3.0 standard deviations. This excess provides evidence for the Higgs boson decay into b-quarks and for its production in association with a vector boson. The combination of this result with that of the Run 1 analysis yields a ratio of the measured signal events to the Standard Model expectation equal to 0.90 ± 0.18(stat.) − 0.19 + 0.21 (syst.). Assuming the Standard Model production cross-section, the results are consistent with the value of the Yukawa coupling to b-quarks in the Standard Model.