The production of $W$ boson pairs in association with one jet in $pp$ collisions at $\sqrt{s} = 8$ TeV is studied using data corresponding to an integrated luminosity of 20.3 fb$^{-1}$ collected by the ATLAS detector during 2012 at the CERN Large Hadron Collider. The cross section is measured in a fiducial phase-space region defined by the presence of exactly one electron and one muon, missing transverse momentum and exactly one jet with a transverse momentum above 25 GeV and a pseudorapidity of $|\eta|<4.5$. The leptons are required to have opposite electric charge and to pass transverse momentum and pseudorapidity requirements. The fiducial cross section is found to be $\sigma^{\mathrm{fid,1\textrm{-}jet}}_{WW}=136\pm6($stat$)\pm14($syst$)\pm3($lumi$)$ fb. In combination with a previous measurement restricted to leptonic final states with no associated jets, the fiducial cross section of $WW$ production with zero or one jet is measured to be $\sigma^{\mathrm{fid,}\leq\mathrm{1\textrm{-}jet}}_{WW}=511\pm9($stat$)\pm26($syst$)\pm10($lumi$)$ fb. The ratio of fiducial cross sections in final states with one and zero jets is determined to be $0.36\pm0.05$. Finally, a total cross section extrapolated from the fiducial measurement of $WW$ production with zero or one associated jet is reported. The measurements are compared to theoretical predictions and found in good agreement.