The Wallabies scored five tries to two but were made to sweat after Bernard Foley missed four straight shots at goal to keep the Pumas in the contest.
Reece Hodge bagged a decisive double at Estadio Malvinas Argentinas, with fellow winger Marika Koroibete, Foley and halfback Will Genia also crossing at Estadio Malvinas Argentinas.
With South Africa earlier on Sunday losing 25-24 to New Zealand in a thriller in Cape Town, Australia’s victory earned the Wallabies second place in the championship behind the unbeaten All Blacks for the second year running.
The Test was eerily similar to the Wallabies’ 45-20 win over the Pumas in Canberra, with the score locked up at 13-all at halftime and 20-all midway through the second stanza.
Foley’s 62nd-minute converted try and penalty shortly after secured the points for Australia after the star playmaker had an uncustomary night with the boot.
Foley had slotted 19 consecutive goals before losing his rhythm on Sunday.
The Wallabies dominated possession and territory but spirited defence and Foley’s errant kicking kept the Pumas in the match until late.
“The Argentinians turned up for a really big arm wrestle, in particular in that first half,” said Wallabies captain Michael Hooper.
“Even up ’til that 60-minute mark there, it was a really hard game to play. It was a hard travel schedule we have so I’m really proud for a young team that we have been able to grow and to get the win here tonight.”
Michael Cheika’s men finished the tournament with two wins, two draws against the Springboks and two losses to New Zealand.
They will rest up after a brutal fortnight in South Africa and South America before getting one last crack at the All Blacks in Brisbane on October 21 in the last Bledisloe Cup clash of the year.