The Eocene-Oligocene Transition (EOT), approximately 34 Ma ago, marks a period of major global cooling and inception of the Antarctic ice sheet. Proxies of deep circulation suggest a contemporaneous onset or strengthening of the Atlantic meridional overturning circulation (AMOC). Proxy evidence of gradual salinification of the North Atlantic and tectonically driven isolation of the Arctic suggest that closing the Arctic-Atlantic gateway could have triggered the AMOC at the EOT. We demonstrate this trigger of the AMOC using a new paleoclimate model with late Eocene boundary conditions. The control simulation reproduces Eocene observations of low Arctic salinities. Subsequent closure of the Arctic-Atlantic gateway triggers the AMOC by blocking freshwater inflow from the Arctic. Salt advection feedbacks then lead to cessation of overturning in the North Pacific. These circulation changes imply major warming of the North Atlantic Ocean, and simultaneous cooling of the North Pacific, but no interhemispheric change in temperatures.