Over the last quarter, Alemanno conceded a penalty try and yellow card, Hodge kicked his penalty and converted tries by Folau Fainga'a — their bonus-point fourth — and Len Ikitau.
Australia's superb record against the Pumas — unbeaten since 2018 and unbeaten in Argentina since 2014 — was in doubt after Hooper withdrew from the squad 24 hours beforehand after revealing his mindset wasn't right.
He was heading home with first-choice hooker Dave Porecki, who was concussed in training.
"We really wanted to put in an effort that our fans can be proud of but also our captain, Hoops, can be proud of," stand-in skipper James Slipper said.
"We were thinking about him all day and we were playing for him tonight."They were already without their best back, Samu Kerevi, then playmaker Quade Cooper limped off in the second half with an apparent recurrence of the calf injury which sidelined him from the England series last month.
But their fill-ins all shone, notably flanker Fraser McReight, who capped his first test start with a try.
The first half was back and forth and scrappy, as both teams took their one and only real try chances.
Argentina was over after only six minutes from lineout ball. An inside pass from srumhalf Tomas Cubelli to wing Emiliano Boffelli opened up the Australians, and another inside pass from flyhalf Santiago Carreras to No. 8 Pablo Matera put him clear and over the tryline.
Boffelli and Cooper traded goalkicks without missing in the half, and midway through the Wallabies scored their try, also from lineout ball.
Cooper delayed his pass for right wing Jordan Petaia to bust pass two defenders on the angle.
But the Wallabies were error-prone and Boffelli kicked Argentina to a 19-10 halftime lead.
As the Wallabies turned the heat up in the second half, Cooper hurt his left calf merely stepping off it and Hodge came in without missing a beat.
McReight, playing his first test in a year, pulled them back after breaking off a lineout maul with a helpful carry from new cap Jed Holloway.
The Pumas restored the nine-point lead when fullback Juan Cruz Mallia counterattacked from deep and Matera sent in fellow back-rower Juan Martin Gonzalez. But that was against the run of play.
Alemanno collapsed another Wallabies lineout maul and paid the price. Hodge then kicked the visitors in front and their confidence was such that, with another close-range penalty, they went for the attacking lineout instead.
The gamble paid off when Fainga'a blew over.Boffelli finally missed a goalkick — from halfway — at his seventh attempt as the beaten Pumas vainly sought a losing bonus point, and a helter-skelter ending finished with Ikitau scoring a converted try.
They move to San Juan next week for round two.
