This Letter presents the observation and measurement of electroweak production of a same-sign W boson pair in association with two jets using 36.1 fb-1 of proton-proton collision data recorded at a center-of-mass energy of s=13 TeV by the ATLAS detector at the Large Hadron Collider. The analysis is performed in the detector fiducial phase-space region, defined by the presence of two same-sign leptons, electron or muon, and at least two jets with a large invariant mass and rapidity difference. A total of 122 candidate events are observed for a background expectation of 69±7 events, corresponding to an observed signal significance of 6.5 standard deviations. The measured fiducial signal cross section is σfid=2.89-0.48+0.51(stat)-0.28+0.29(syst) fb.