Then stand-in halfback Nick Phipps turned hero by darting to the blind and bursting past two defenders to put Digby Ioane over.
A cramping Berrick Barnes, who missed two easy first-half penalties, nailed the wide conversion and Australia were in the lead with 10 minutes left.
Kurtley Beale, after coming off the bench, landed a monster penalty with a minute left and the gallant Pumas were denied an historic first Rugby Championship victory.
The win moved the Wallabies (8) into second place on the tournament standings - eight points behind New Zealand (16) who have a vice-like grip on the series with just two rounds to play after a 21-11 win over South Africa.
Argentina had scared both the All Blacks and the Springboks in their opening three games but no more than the mighty fright they gave the mostly willing but off-key Wallabies outfit.
Barnes had just put the Wallabies back on level pegging at 6-all in the 49th minute when the Test took a shock turn in a crazy three minutes of madness.
Cooper was charged down following the restart by replacement flanker Tomas Leonardi, who scored with his first touch of the ball.
Then stand-out playmaker Juan Martin Hernandez produced a brilliant kick-return from a poorly-chased Barnes high kick that saw Juan Imhoff ultimately put Julio Farais Cabello over for a 13-point lead.
It was a try dripping in controversy as Imhoff put his right foot into touch in the lead-up to the chagrin of most in the 22,278-strong Gold Coast crowd.
Hernandez kicked his third penalty of the match soon after and it looked like Australia were headed for another major boilover loss to go with the defeats to Scotland, Samoa and Ireland over the past 15 months.
The Wallabies had showed their intention to attack from the outset, spinning the ball wide and recycling the ball rapidly to pose plenty of questions of the Pumas' defence.
Five-eighth Cooper had a forgettable first half, fumbling several times and also throwing an early intercept pass, but did also have his moments and put Sharpe through a gap on halftime only for the captain to be held up.
Deans admitted he was considering replacing Cooper but held off largely due to winger Dom Shipperley nursing a dislocated finger.
"It did cross our mind. We were to-ing and fro-ing there for a while but Dom's injury did complicate things ... and could leave (the bench) exposed," he said.
Winger Ioane was named man-of-the-match for his energetic display, Barnes made a fine fist of his first Test at fullback and young ball-scavenger Michael Hooper again stood out.
No.8 Radike Samo produced a powerful first half but did botch a try by dropping the ball in the corner while McCabe bounced back strongly in his return from injury despite being harshly sin-binned after running down flying Pumas winger Gonzalo Camacho.
Disappointed Pumas coach Santiago Phelan and skipper Juan Martin Fernandez Lobbe were proud of their performance but felt they let a major opportunity slip.
"In the last 20 minutes, we tried to hold it instead of trying the grab the game," No.8 Lobbe said.
- AAP
