This Letter presents the first study of W^±W^±jj, same-electric-charge diboson production in association with two jets, using 20.3 fb⁻¹ of proton-proton collision data at √s̅ = 8 TeV recorded by the ATLAS detector at the Large Hadron Collider. Events with two reconstructed same-charge leptons (e^±e^±, e^±μ^±, and μ^±μ^±) and two or more jets are analyzed. Production cross sections are measured in two fiducial regions, with different sensitivities to the electroweak and strong production mechanisms. First evidence for W^±W^±jj production and electroweak-only W^±W^±jj production is observed with a significance of 4.5 and 3.6 standard deviations, respectively. The measured production cross sections are in agreement with standard model predictions. Limits at 95% confidence level are set on anomalous quartic gauge couplings.