This Letter reports the observation of W+W−γ triboson production in 140 fb−1 of data collected by the ATLAS detector from proton–proton collisions at a centre-of-mass energy of s√ = 13 TeV at the LHC. Events with an opposite-charge eμ pair, a high transverse-momentum photon, and significant missing transverse momentum are considered. The observed (expected) significance of the signal is 5.9 (6.0) standard deviations. The measured fiducial cross-section, defined for the W+W−γ→e±μ∓νν¯γ final state is 6.2  ±  0.8 (stat.)  ±  0.6 (sys.) fb, in good agreement with the Standard Model prediction of 6.1+1.0−0.7 fb. Constraints on the Wilson coefficients of 13 dimension-8 operators describing physics beyond the Standard Model through anomalous quartic gauge-boson couplings are derived using the effective field theory framework.