This letter reports the observation of photon-induced production of W-boson pairs, gamma gamma -> WW. The analysis uses 139 fb(-1) of LHC proton-proton collision data taken at root 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(+/-)nu mu(-/+)nu 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 gamma gamma -> 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.