Winger Digby Ioane scored a late try for the Wallabies whose embattled coach Robbie Deans was vindicated for selecting a heavy set of forwards to take on Argentina's might up front following a string of injuries before the match.
Full-back Michael Harris kicked 20 points, including five first-half penalties to put Australia 15-9 ahead at half-time as the Wallabies made the fewer handling mistakes with a slippery ball in damp conditions.
Australia's third victory in the elite southern hemisphere tournament saw them finish equal with South Africa in points, but secured the runner-up position by virtue of having one more win than the Springboks.
The world champions All Blacks crushed South Africa earlier on Saturday for a clean sweep of six wins, while Argentina had to settle for bottom place but enjoyed an encouraging debut with a draw and five mostly competitive defeats.
"The character shown and the intensity was outstanding tonight and that provided the platform for our victory," Australia captain Nathan Sharpe said.
"On a field that was smaller than we're used to and one of the wettest balls we've played with and, in my career, one of the most hostile crowds, we got the job done."
Australia struck their telling blow with Ioane's try with 15 minutes remaining, right after surviving Argentina's most dangerous attacking onslaught.
From a scrum on the Pumas' 22-metre line, the ball came out to Ioane at midfield and the winger cut through a gap in the defence to touch down under the posts with Harris converting.
Replacement winger Juan Imhoff scored a late try for the Pumas which was converted by centre Marcelo Bosch.
"The (Argentina) team really gave their all. There were some mistakes, especially in the first half, which finally cost us the match," Pumas winger Gonzalo Camacho said.
"We gave away a lot of penalties, which was something we had set out to avoid, and the Australians got most of their points that way," he said.
The first half was dominated by penalties with Harris slotting over his five while Argentina fly-half Juan Martin Hernandez converted three before missing two that would have made the score level at half-time.
Argentina lost lock Patricio Albacete to the sin bin in the 26th minute but the only advantage Australia were able to take was Harris's fifth penalty.
Harris had another penalty attempt in the 55th minute but missed. A laser beam that had appeared on the field during the first half appeared to be trained on him for some seconds as he set up the ball.
After a barren 23 minutes of play in the second half, Bosch reduced Australia's lead to three points with a penalty after the Pumas had halted an exciting solo break by flanker Michael Hooper at their 22-metre line.
The final 15 minutes saw a flurry of action with wing Horacio Agulla shunted into touch at the last gasp before Ioane scored his try.
Imhoff gave the Pumas hope of a last-gasp win by running over for Argentina in a maul, but Australia coolly closed out victory under great pressure, down to 14 men in the late stages after replacement back Brett Sheehan was sent off for a rough tackle.
Pumas prop Rodrigo Roncero, a veteran of three World Cups, came off to a standing ovation four minutes from the end in a fitting farewell in his 55th and last international.
"I don't know how we'll manage without him, how we'll fill the hole he leaves. He's a master, there are no words to describe him," Camacho said.