This paper describes a measurement of fiducial and differential cross sections of gluon-fusion Higgs boson production in the $H\rightarrow W W^\ast \rightarrow e\nu\mu\nu$ channel, using 20.3 fb$^{-1}$ of proton-proton collision data. The data were produced at a centre-of-mass energy of $\sqrt{s} = 8$ TeV at the CERN Large Hadron Collider and recorded by the ATLAS detector in 2012. Cross sections are measured from the observed $H\rightarrow W W^\ast \rightarrow e\nu\mu\nu$ 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.