Higgs boson production cross-sections in proton-proton collisions are measured in the $H \to WW^{\ast} \to e\nu\mu\nu$ decay channel. The proton-proton collision data were produced at the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector in 2015 and 2016, corresponding to an integrated luminosity of 36.1 fb$^{-1}$. The product of the $H \to WW^{\ast}$ branching fraction times the gluon-gluon fusion and vector-boson fusion cross-sections are measured to be $11.4 ^{+1.2}_{-1.1} (\mathrm{stat.}) ^{+1.8}_{-1.7}(\mathrm{syst.})$ $\mathrm{pb}$ and $0.50 ^{+0.24}_{-0.22} (\mathrm{stat.}) \pm 0.17(\mathrm{syst.})$ $\mathrm{pb}$, respectively, in agreement with the Standard Model predictions.