Approximately 34 million years ago across the Eocene-Oligocene transition (EOT), Earth's climate tipped from a largely unglaciated state into one that sustained large ice sheets on Antarctica. Antarctic glaciation is attributed to a threshold response to slow decline in atmospheric CO2 but our understanding of the feedback processes triggered and of climate change on the other contents is limited. Here we present new geochemical records of terrigenous dust accumulating on the sea floor across the EOT from a site in the central equatorial Pacific. We report a change in dust chemistry from an Asian affinity to a Central-South American provenance that occurs geologically synchronously with the initiation of stepwise global cooling, glaciation of Antarctica and aridification on the northern continents. We infer that the inter-tropical convergence zone of intense precipitation extended to our site during late Eocene, at least four degrees latitude further south than today, but that it migrated northwards in step with global cooling and initiation of Antarctic glaciation. Our findings point to an atmospheric teleconnection between extratropical cooling and rainfall climate in the tropics and the mid-latitude belt of the westerlies operating across the most pivotal transition in climate state of the Cenozoic Era.