This paper reports the observation of electroweak diboson (WW/WZ/ZZ) production in association with a high-mass dijet system, in which final states with one boson decaying leptonically and the other boson decaying hadronically are studied. The hadronically decaying W/Z boson is reconstructed as either two small-radius jets or one large-radius jet with jet substructure requirements. The data analyzed correspond to an integrated luminosity of 140 fb−1 of proton–proton collisions at a center-of-mass energy of s√=13 TeV collected with the ATLAS detector during the 2015-2018 data taking at the Large Hadron Collider. The electroweak production of WW/WZ/ZZ in association with two jets is observed in a phase space dominated by vector-boson scattering with a significance of 7.4σ (expected 6.1σ) and the signal strength is determined to be 1.28+0.23−0.21. The corresponding production cross section in a fiducial phase space is measured in addition. The signal strengths of both electroweak and QCD associated diboson productions are furthermore measured in a two-dimensional fit, the result of which agrees with the Standard Model prediction. The data are interpreted in the context of a dimension-8 effective field theory to probe anomalous quartic gauge couplings resulting in the first set of exclusion limits on the Wilson coefficients in the semileptonic channel reported by the ATLAS Collaboration. The observed limits for the S02, T0 and M0 operators are (−3.96<fS02/Λ4<3.96) TeV−4, (−0.25<fT0/Λ4<0.22) TeV−4, (−1.26<fM0/Λ4<1.25) TeV−4.