Ireland dominated possession in the second half but was never able to gain the lead.
It was Australia's second one-point victory over Ireland in a World Cup match, after the classic 19-18 quarter-final victory at Lansdowne Road 12 years ago.
Australia now plays Scotland in a quarter-final in Brisbane next weekend, while Ireland will be back at Telstra Dome on November 9 to take on France.
Australian skipper George Gregan opened the scoring in the 10th minute with only the third drop goal of his 92-Test career.
Rampaging Wallabies backrower George Smith then scored the only try of the opening half three minutes later.
Winger Wendell Sailor made two charges at the Irish line before Australia swung the ball left, where Smith got on the end of an overlap to score in the corner.,p>Ireland's only scores of the first half were two penalty goals to five-eighth Ronan O'Gara while Elton Flatley slotted one for the Wallabies, who led 11-6 at the interval.
Both teams lost a player to the sin-bin in the dying minutes of the first half. Australian fullback Mat Rogers was carded for slowing the ball in the ruck, while Irish winger Shane Horgan joined him on the sidelines for excessive rucking.,p>Flying centre Brian O'Driscoll had the large Irish contingent in the capacity crowd of 54,206 in raptures when he scored Ireland's only try in the 50th minute, powering over to score in the left corner.
O'Gara's successful conversion brought Ireland within a point, only for Flatley to increase Australia's advantage to 17-13 with a penalty shortly afterwards.
Ireland declined a couple of kickable penalty goal attempts but was unable to breach the Australian line. O'Driscoll then brought Ireland within a point of the lead for the second time with a 68th minute drop goal.
Replacement David Humphreys missed with another drop goal attempt seven minutes later that would have given Ireland the lead and the heavily favoured Australians were able to hold on for the narrowest of victories.
(Report - AAP for Rugby Heaven).