Measurements of the W±→ℓ±ν and Z→ℓ+ℓ− production cross sections (where ℓ±=e±,μ±) in proton–proton collisions at s=13 TeV are presented using data recorded by the ATLAS experiment at the Large Hadron Collider, corresponding to a total integrated luminosity of 81 pb−1. The total inclusive W±-boson production cross sections times the single-lepton-flavour branching ratios are σW+tot=11.83±0.02 (stat)±0.32 (sys)±0.25 (lumi) nb and σW−tot=8.79±0.02 (stat)±0.24 (sys)±0.18 (lumi) nb for W+ and W−, respectively. The total inclusive Z-boson production cross section times leptonic branching ratio, within the invariant mass window 66<mℓℓ<116 GeV, is σZtot=1.981±0.007 (stat)±0.038 (sys)±0.042 (lumi) nb. The W+, W−, and Z-boson production cross sections and cross-section ratios within a fiducial region defined by the detector acceptance are also measured. The cross-section ratios benefit from significant cancellation of experimental uncertainties, resulting in σW+fid/σW−fid=1.295±0.003 (stat)±0.010 (sys) and σW±fid/σZfid=10.31±0.04 (stat)±0.20 (sys). Theoretical predictions, based on calculations accurate to next-to-next-to-leading order for quantum chromodynamics and to next-to-leading order for electroweak processes and which employ different parton distribution function sets, are compared to these measurements.