The Republic went a goal behind on 49 minutes, scored by Leeds striker Mark Viduka and assisted by Blues winger Stan Lazaridis., which was equalised by Manchester United's John O'Shea after 74 minutes.
Enter Morrison, who within 10 seconds of appearing after 57 minutes had flashed a powerful left-footed effort inches wide and moments later missed a glorious chance when completely missing a header after a great cross from Ian Harte.
On 80 minutes however, Morrison scored what proved to be the winner for the Irish, running on to a long ball over the top of the Socceroo's defence he calmly trickled the ball past the on-rushing keeper, Mark Schwarzer and into an empty net.
Kenny Cunningham skippered the Irish and played for 84 minutes, while Lazaridis played the full ninety for the Socceroos.
Republic of Ireland: Colgan, Carr, Breen, Cunningham, O'Shea, Holland, Kinsella, Finnan, Duff, Keane, Doherty. Subs: O'Brien, Dunne, Harte, Healy, Murphy, Quinn, Kilbane, Connolly, Morrison, Given.
Australia: Schwarzer, Neill, Popovic, Lazaridis, Emerton, Okon, Bresciano, Tiatto, Viduka, Chipperfield, Muscat. Subs: Vidmar, Skoko, Aloisi, Grella, Kalac, Foxe, Wilkshire.