This paper describes a measurement of fiducial and differential cross sections of gluon-fusion Higgs boson production in the H → W W∗→ eνμν channel, using 20.3 fb−1 of proton-proton collision data. The data were produced at a centre-of-mass energy of s=8 TeV at the CERN Large Hadron Collider and recorded by the ATLAS detector in 2012. Cross sections are measured from the observed H→ W W∗→ eνμν signal yield in categories distinguished by the number of associated jets. The total cross section is measured in a fiducial region defined by the kinematic properties of the charged leptons and neutrinos. Differential cross sections are reported as a function of the number of jets, the Higgs boson transverse momentum, the dilepton rapidity, and the transverse momentum of the leading jet. The jet-veto efficiency, or fraction of events with no jets above a given transverse momentum threshold, is also reported. All measurements are compared to QCD predictions from Monte Carlo generators and fixed-order calculations, and are in agreement with the Standard Model predictions.[Figure not available: see fulltext.]