The hosts had the opportunity to seal victory with a last-gasp penalty but Reece Hodge could not get his effort on target.
After Hodge kicked Australia into the lead in the 16th minute following a yellow card for Argentina’s Marcos Kremer, penalties from Nicolas Sanchez and – with Wallabies skipper Michael Hooper then sin-binned – Domingo Miotti in reply put the Pumas 6-3 up.
Bautista Delguy subsequently registered the first try of the contest, following good work from Felipe Ezcurra, to make it 13-3.
Hodge kicks either side of the break took the score to 13-9 before Australia replacement Lukhan Salakaia-Loto was red-carded for a tackle on Santiago Grondona and Miotti converted the resulting penalty.
A yellow for Lucas Paulos then made it 14 men apiece, and moments later Hooper crossed to bring the score level. Hodge then failed to take his winning chance as he sent his 80th-minute kick wide of the posts.
Argentina – who were without regular captain Pablo Matera, Guido Petti and Santiago Socino despite their suspensions over historic posts on social media having been lifted by the Argentine Rugby Union on Thursday – finish second in the table, with Australia third, both on eight points from four matches.
New Zealand took the trophy having already all but secured it, with Argentina and Australia having required an unrealistically large winning margin in Sydney to replace the All Blacks at the top of the standings.