Hodge gave Australia a 16th-minute 3-0 lead after Marco Kremer was harshly sin-binned for contact to the head.
With that precedent set, Hooper was also sidelined for 10 minutes with a similarly contentious call after half an hour.
The Pumas capitalised, taking the lead with a second penalty and the game's first try. Their pack rumbled the ball forward from a lineout deep in their own half and Felipe Ezcurra broke free to set up wing Bautista Delguy for a well-taken try.
That took the visitors to a 13-3 lead despite the Wallabies enjoying around 70 per cent territory and possession in the first half.
The hosts pressured the Pumas line right on halftime, ultimately settling for a 43rd-minute penalty to close the gap to 13-6 at the break.
Hodge kicked his third penalty for 13-9 but the match turned firmly Argentina's way when Salakaia-Loto was sent off on the hour for a high shot on Santiago Grondona.
Unlike the first-half yellow cards, the dismissal of Salakaia-Loto — on the pitch for just nine minutes — was justified and Argentina went 16-9 ahead from the resultant penalty.
But the Pumas enjoyed a one-man advantage for just six minutes, with Lucas Paulos sin-binned for collapsing a maul. Australian persistence was rewarded with Hooper's try and Hodge's conversion for 16-16, but once again, the Wallabies were unable to break the deadlock.