Higgs boson production via gluon–gluon fusion is measured in the $WW^{*} \rightarrow e\nu \mu \nu $ decay channel. The dataset utilized corresponds to an integrated luminosity of 139 fb$^{-1}$ collected by the ATLAS detector from $\sqrt{s}=13$ TeV proton–proton collisions delivered by the Large Hadron Collider between 2015 and 2018. Differential cross sections are measured in a fiducial phase space restricted to the production of at most one additional jet. The results are consistent with Standard Model expectations, derived using different Monte Carlo generators.