The Wallabies will take series honours after their 27-21 win in San Salvador de Jujuy last week, and remain unbeaten in four matches under new coach Les Kiss, but will rue not finishing off a shorthanded Pumas side.
Matera grabbed his second try to put the home side ahead for the first time in the 64th minute but calamity struck for the Pumas six minutes later.
Lucio Cinti was shown a yellow card for a deliberate knock-on and, from the resulting penalty, referee James Doleman sent Santiago Carreras to join him in the sin bin for the same infraction while awarding Australia a penalty try to level up the scores.
Australia's backline has not really fired yet under Kiss and that lack of fluency, along with a tendency towards error, cost them dearly as they chased the winning score.
"It's difficult to come here and do what we did," said Kiss.
"But we know we let them off the hook. We took control, then let it go. Took control again, then let it go again.
"Series win, we get a trophy, but we know there's plenty we need to work on."
The Wallabies had started the match well and held a 14-0 lead after scoring two tries inside half an hour at a cool and windy Estadio Malvinas Argentinas.
Shortly after having a try called back for a foot in touch, recalled Australia captain Harry Wilson opened the scoring from close range in the 20th minute.
Pumas lock Franco Molina was sent to the sin bin in the 27th minute and the visitors quickly took advantage of the extra man to send winger Max Jorgensen across the line after the forwards had drawn in the defence.
Scrumhalf Ryan Lonergan converted both tries but the visitors were well aware they had held a similarly comfortable lead in the second test on their 2024 tour only to lose 67-27.
The Pumas duly hit back with two tries in four minutes before halftime after a yellow card shown to Australia lock Jeremy Williams evened up the numbers.
Matera crashed over under the posts before winger Ignacio Mendy flashed through the Australian defensive line in a well-worked move from a lineout.
Fullback Santiago Carreras slotted the conversions to level up the scores and Argentina could have gone into halftime with the lead had they kicked either of the penalties they were awarded on the Australia line just before the break.
Flanker Fraser McReight forced his way over the line to put Australia back ahead early in the second half but a similar converted try from Argentina hooker Ignacio Ruiz made it 21-21 with 56 minutes on the clock.