Electroweak symmetry breaking explains the origin of the masses of elementary particles through their interactions with the Higgs field. Besides the measurements of the Higgs boson properties, the study of the scattering of massive vector bosons with spin one allows the nature of electroweak symmetry breaking to be probed. Among all processes related to vector-boson scattering, the electroweak production of two jets and a $Z$-boson pair is a rare and important one. Here we report the observation of this process from proton-proton collision data corresponding to an integrated luminosity of 139/fb recorded at a centre-of-mass energy of 13 TeV with the ATLAS detector at the Large Hadron Collider. We consider two different final states originating from the decays of the $Z$-boson pair - one containing four charged leptons and the other containing two charged leptons and two neutrinos. The hypothesis of no electroweak production is rejected with a statistical significance of 5.7 $\sigma$, and the measured cross-section for electroweak production is consistent with the standard model prediction. In addition, we report cross-sections for inclusive production of a $Z$-boson pair and two jets for the two final states.