Montoya’s men came into the clash on the back of last weekend’s gut-wrenching after-the-siren defeat in Townsville, where they squandered a 14-point halftime advantage to crash 28-24.
After some late, nervy moments, they held on this time at the Sydney Football Stadium to dent the Wallabies’ chances of claiming a first southern hemisphere title in a decade.
Both sides have now won two and lost two, but Argentina will take heart from a rare win on the road, having previously lost five of their last six away tests, including three on the trot.
The Wallabies must now regroup before back-to-back tests against New Zealand, in Auckland and Perth, while the Pumas face a daunting double-header against world champions South Africa.
Both sides came out swinging in front of a sellout crowd with two tries in the opening seven minutes.
Gloucester first five-eighths Carreras nailed a penalty to put Argentina ahead, but Australia responded immediately with the power of Joseph-Aukuso Suaalii carrying him over for a converted try.
But minutes later there was a horror moment for Australia when first five-eighths Tane Edmed, making his starting debut, had his kick on the Wallabies’ 22 charged down by Montoya, who collected and slid over the line.
Carreras added the extras and then another penalty to put Argentina 13-7 ahead after a frantic first 15 minutes, with a yellow card for wing Max Jorgenson after a deliberate knock-on piling more pressure on the hosts.
With Australia committing too many errors, Carreras landed two more penalties as the Pumas marched to halftime in control with only good defence preventing the Wallabies trailing by more.
In Townsville, the Wallabies had been a different team in the second half, but they got off to a poor start with another sloppy play allowing Carreras to slot his fifth penalty.
More lapses in concentration, this time from Billy Pollard and Harry Wilson, allowed Carreras to bag his sixth and seventh of the day, but to Australia’s relief he was then sin-binned.
In his absence, and roared on by the crowd, Australia scored two tries in two frantic minutes with Andrew Kellaway and Filipo Daugunu crossing to reduce the deficit to 28-19 with 14 minutes left.
It set up a blockbuster finish, with Daugunu getting another in the corner as the Pumas tired and the gap closed to 28-26 with a few seconds left on the clock.
But after going through several phases after the siren, Australia coughed up a final penalty enabling Argentina to kick to touch and celebrate victory.