Abstract Higgs boson production via gluon–gluon fusion is measured in the $$WW^{*} \rightarrow e\nu \mu \nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> <mml:mo>→</mml:mo> <mml:mi>e</mml:mi> <mml:mi>ν</mml:mi> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:math> decay channel. The dataset utilized corresponds to an integrated luminosity of 139 fb $$^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> collected by the ATLAS detector from $$\sqrt{s}=13$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> <mml:mo>=</mml:mo> <mml:mn>13</mml:mn> </mml:mrow> </mml:math> 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.