A search for the decay of the Standard Model Higgs boson into a $b\bar{b}$ pair when produced in association with a $W$ or $Z$ boson is performed with the ATLAS detector. The data, corresponding to an integrated luminosity of 79.8 $\mathrm{fb}^{-1}$ were collected in proton-proton collisions during Run 2 of the Large Hadron Collider at a centre-of-mass energy of 13 TeV. 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 (expected) significance of 4.9 (4.3) standard deviations. A combination with the results from other searches in Run 1 and in Run 2 for the Higgs boson in the $b\bar{b}$ decay mode is performed, which yields an observed (expected) significance of 5.4 (5.5) standard deviations, thus providing direct observation of the Higgs boson decay into $b$-quarks. The ratio of the measured event yield for a Higgs boson decaying into $b\bar{b}$ to the Standard Model expectation is $1.01 \pm 0.12 \mathrm{(stat.)} ^{+0.16}_{-0.15} \mathrm{(syst.)}$. Additionally, a combination of Run 2 results searching for the Higgs boson produced in association with a vector boson yields an observed (expected) significance of 5.3 (4.8) standard deviations.