With Argentina having held South Africa in Mendoza in August, there was never any doubt the Wallabies faced a tough task, and with injuries piling up and rumblings of discontent in the camp the preparation was far from ideal.
But coach Robbie Deans saw his team dig deep, their one try coming from Digby Ioane in the 65th minute, converted by Harris who missed just one kick all night.
Argentina came back at their visitors and guaranteed a tense finish when Juan Jose Imhoff was driven over the line, but Australia clung on.
For Argentina, Juan Martin Hernandez kicked three penalties in the first half to five from Harris.
Hernandez had to come off injured five minutes into the second half and Marcelo Bosch stepped up to kicking duties, slotting a penalty and adding the extras to Imhoff's try.
That try came four minutes after veteran prop Rodrigo Roncero bowed out of international rugby, substituted in the 73rd minute in his final Test.
Roncero has been a cornerstone of Argentinian rugby for 14 years and has appeared in three Rugby World Cups, helping the Pumas to their third-place finish in 2007.
The 35-year-old knew that signing off with a win against the Wallabies on his 55th appearance would have been the crowning glory of his career, but he was unable to play a part in the closing stages.
Australia's success earned them a second-placed finish, behind world champions New Zealand who won all six of their matches, in the inaugural Rugby Championship. Argentina finished bottom, with five losses and their one draw against the Springboks.
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