Tries from Iain Henderson and Garry Ringrose helped the hosts to a 17-0 lead but the Wallabies fought back with three touchdowns to move into the lead.
Keith Earls' score proved decisive but it was the Irish forwards who laid the platform in a breathless encounter.
The result ended Australia's hopes of a Grand Slam over the home nations.
They had defeated Wales, Scotland and France and will face England in the final match of their northern hemisphere tour next week.
