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 sqrt[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 σ^{fid}=2.89_{-0.48}^{+0.51}(stat)_{-0.28}^{+0.29}(syst)  fb.