An observation is reported of the electroweak production of a W⁺W^- pair in association with two jets, with both W bosons decaying leptonically. The data sample corresponds to an integrated luminosity of 138 fb^-1 of proton-proton collisions at √s = 13 TeV, collected by the CMS detector at the CERN LHC. Events are selected by requiring exactly two opposite-sign leptons (electrons or muons) and two jets with large pseudorapidity separation and high dijet invariant mass. Events are categorized based on the flavor of the final-state leptons. A signal is observed with a significance of 5.6 standard deviations (5.2 expected) with respect to the background-only hypothesis. The measured fiducial cross section is 10.2±2.0fb and this value is consistent with the standard model prediction of 9.1±0.6fb.