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The Wallabies outscored Ireland by four tries to nil to record a comfortable 32-15 victory in Dublin on Saturday.
The opening minutes were scrappy but the visitors broke the deadlock when Quade Cooper landed a penalty from long range in the ninth minute.
Ireland struck back through Jonathan Sexton four minutes later after good Irish pressure provoked an Australian infringement.
Cooper then displayed the two sides of his play, missing a very kickebable penalty and then in their next attack looping around and delivering a great sleight-of-hand pass to Stephen Moore who offloaded to Nick Cummins.
The 26-year-old wing stepped inside fullback Rob Kearney to run in and touch down for his fifth international try, Cooper adding the conversion for 10-3.
Sexton subsequently missed an easy penalty and Brian O'Driscoll was taken off to the blood bin and replaced by Robbie Henshaw.
Henshaw, however, missed two tackles in the lead-up to the Aussies' second try, with Cooper and Moore once again prominent before flank Scott Fardy produced a great offload to Hooper, who ran in unopposed for his first try for his country in his 22nd appearance.
Cooper's kicking was, fortunately for the Irish, not as accurate as his passing and he dragged his conversion wide to leave the score 15-3 after 25 minutes.
Sexton reduced the deficit with a penalty in front of the posts on the half hour mark.
The Irish had recovered their composure and a stunning break by Fergus McFadden got them deep into Australian territory and eventually won a penalty with Hooper sent to the sin bin by referee Chris Pollock.
Sexton slotted it over for 15-9and the flyhalf added another just before half-time to put Ireland right back in the game despite having conceded two tries but the half ended on a low note for the kicker as he seemingly pulled up with a hamstring injury.
Sexton as feared did not return to the fray after the break with Ian Madigan, selected ahead of Paddy Jackson because of his versatility, given the opportunity to impress.
The Wallabies didn't have to wait long for their third try as Cooper produced an individual moment of magic, feinting to pass before stepping inside Luke Marshall to touch down, and converting it for 22-12.
Cooper, rewarded by coach Ewen McKenzie for his change in attitude by being named vice-captain for the tour, then added a penalty from long range to leave Ireland in danger of a real hammering as the Wallabies led 25-12 with half-an-hour remaining.
Madigan landed a penalty shortly before the hour to just about keep the Irish in touch at 25-15 but the hosts were lacking any sort of spark or inspiration to suggest they could claw it back.
The Irish eschewed the possibility of a relatively easy three points to go for touch with 20 minutes remaining but it went to waste as the Australians once again forced a turnover, their ninth of the match to the hosts' two.
The Wallabies added a fourth try through Hooper after the Australian pack pushed the Irish back over their line, Cooper adding the conversion for 32-15.
The visitors' superb day ended on a sour note when Tevita Kuridrani speartackled Peter O'Mahony and Pollock wielded the red card after consulting the television match official.
Man of the match: Apart from his two tries and yellow card, Michael Hooper had a major influence on the match with his work at the breakdown and work rate making him stand out.
Moment of the match: Quade Cooper's dummy to go over for his try was a great moment of individual brilliance.
Villain of the match: Tevita Kuridrani for his unnecessary tip tackle.
Pens: Sexton 4, Madigan
Tries: Cummins, Hooper 2, Cooper
Cons: Cooper 3
Pens: Cooper 2
Yellow card: Michael Hooper (Australia, 33 - repeated infringements)
Red card: Tevita Kuridrani (Australia, 72 - dangerous tackle)
Ireland: 15 Rob Kearney, 14 Tommy Bowe, 13 Brian O'Driscoll, 12 Luke Marshall, 11 Fergus McFadden, 10 Jonathan Sexton, 9 Eoin Reddan, 8 Jamie Heaslip, 7 Sean O'Brien, 6 Peter O'Mahony, 5 Paul O'Connell (captain), 4 Devin Toner, 3 Mike Ross, 2 Rory Best, 1 Cian Healy.
Replacements: 16 Sean Cronin, 17 Jack McGrath, 18 Stephen Archer, 19 Mike McCarthy, 20 Kevin McLaughlin, 21 Conor Murray, 22 Ian Madigan, 23 Rob Henshaw.
Australia: 15 Israel Folau, 14 Adam Ashley-Cooper, 13 Tevita Kuridrani, 12 Matt Toomua, 11 Nick Cummins, 10 Quade Cooper, 9 Will Genia, 8 Ben Mowen (captain), 7 Michael Hooper, 6 Scott Fardy, 5 James Horwill, 4 Rob Simmons, 3 Sekope Kepu, 2 Stephen Moore, 1 James Slipper.
Replacements: 16 Tatafu Polota-Nau, 17 Benn Robinson, 18 Paddy Ryan, 19 Sitaleki Timani, 20 Liam Gill, 21 Nic White, 22 Christian Leali'ifano, 23 Joe Tomane.
Referee: Chris Pollock (New Zealand)
Assistant referees: Romain Poite (France), Stuart Berry (South Africa)
TMO: Geoff Warren (England)