The climate impact of ocean gateway openings during the Eocene‐Oligocene transition is still under debate. Previous model studies employed grid resolutions at which the impact of mesoscale eddies has to be parameterized. We present results of a state‐of‐the‐art eddy‐resolving global ocean model with a closed Drake Passage and compare with results of the same model at noneddying resolution. An analysis of the pathways of heat by decomposing the meridional heat transport into eddy, horizontal, and overturning circulation components indicates that the model behavior on the large scale is qualitatively similar at both resolutions. Closing Drake Passage induces (i) sea surface warming around Antarctica due to equatorward expansion of the subpolar gyres, (ii) the collapse of the overturning circulation related to North Atlantic Deep Water formation leading to surface cooling in the North Atlantic, and (iii) significant equatorward eddy heat transport near Antarctica. However, quantitative details significantly depend on the chosen resolution. The warming around Antarctica is substantially larger for the noneddying configuration (∼5.5°C) than for the eddying configuration (∼2.5°C). This is a consequence of the subpolar mean flow which partitions differently into gyres and circumpolar current at different resolutions. We conclude that for a deciphering of the different mechanisms active in Eocene‐Oligocene climate change detailed analyses of the pathways of heat in the different climate subsystems are crucial in order to clearly identify the physical processes actually at work.