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 well above 5 standard deviations consistent 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. This is found to be in agreement with the Standard Model prediction.