Cooper was handed a yellow card after collecting Argentina's flyhalf Nicolas Sanchez around the head off the play.
Wallabies back-rower David Pocock was subbed off in the 36th minute, but it was initially unclear whether it was an injury-related move, or merely tactical.
The Wallabies scored two early tries in their 29-15 win over Argentina in last year's World Cup semi-final.
And it was a case of deja vu in Saturday's re-match as the Wallabies went on an early rampage.
Outside centre Samu Kerevi started the onslaught, touching down after just 43 seconds following a scuffed clearance kick from Argentina.
Hometown hero Dane Haylett-Petty was the next to score following a well-worked lineout and a nice inside pass from Cooper.
And the score was 21-0 after 12 minutes following some magic from Genia.
He unleashed a heavy tackle to knock the ball loose from Sanchez, before soccering the ball forward. The Wallabies' scrumhalf outsprinted a Pumas defender to collect the ball and crash over the line.
Australian prop Scott Sio was sin-binned in the 33rd minute for foul play, but the Wallabies still entered halftime with a 21-6 lead.
However, there were a few nerves when Pumas winger Santiago Cordero scored in the 44th minute. With the margin cut back to eight points, a boilover loomed.
But a barnstorming run from Sean McMahon changed all that.
McMahon bounced off four defenders to set up Genia for his second try in the 52nd minute.
And when Cooper unleashed a no-look inside ball to set up Michael Hooper's try in the 64th minute, the margin had blown out to 20 points and it was as good as game over.
The Wallabies have won two games on the trot after beating South Africa 23-17 last week in Brisbane.
Australia 36 (Will Genia 2, Dane Haylett-Petty, Michael Hooper, Samu Kerevi tries; Bernard Foley 4 conversions, Reece Hodge penalty)
Argentina 20 (Santiago Cordero, Facundo Isa tries; Nicolas Sanchez 2 conversions, 2 penalties).
HT: 21-6.
