This letter reports the observation of photon-induced production of W-boson pairs, γγ→WW. The analysis uses 139 fb−1 of LHC proton-proton collision data taken at s√=13 TeV recorded by the ATLAS experiment during the years 2015-2018. The measurement is performed selecting one electron and one muon, corresponding to the decay of the diboson system as WW→e±νμ∓ν final state. The background-only hypothesis is rejected with a significance of 8.4 standard deviations which is consistent within uncertainty with the expectation from Monte Carlo simulation. A cross section for the γγ→WW process of 3.13±0.31(stat.)±0.28(syst.) fb is measured in a fiducial volume close to the acceptance of the detector, by requiring an electron and a muon of opposite signs with large dilepton transverse momentum and exactly zero additional charged particles.