The Pumas looked to be on their way to a first victory in their maiden season in the southern hemisphere's international championship after tries from Tomas Leonardi and Julio Farias Cabello inside two minutes early in the second half.
The error-prone Wallabies, however, dug deep and battled back with centre McCabe finally breaching the Argentine defence on the hour mark and winger Ioane following suit after a break from scrumhalf Nick Phipps nine minutes later.
"Credit to them, they really put us under a lot of pressure at 19-6 but there is a great deal of courage and commitment to come back ... and finish so strongly," skipper Nathan Sharpe said.
"We're not happy with the close game but there was some good commitment from the guys."
The victory put the Wallabies second in the championship standings with eight points from four matches behind runaway leaders and world champions New Zealand, who have secured 16 points after winning all four of their matches.
Argentina remained bottom of the standings with three points but, along with the 16-16 draw with the Springboks and an encouraging performance against the All Blacks, the match was further evidence they will be competitive in the championship.
"It was close, we're very disappointed," said captain Juan Martin Fernandez Lobbe.
"We got a good lead but this has to be another step forward. We need to try and learn how to close these types of games. These types of teams don't stop until the last minute. That's why we are disappointed, but we must keep going forward.
"We thought we had it, but they're a really good team."
The Wallabies, who also came from behind to beat the Springboks last week after two defeats to the All Blacks to open their campaign, went 3-0 down to a Juan Martin Hernandez penalty in the third minute and will be happy to forget much of the next 20.
Berrick Barnes hit the right upright with his first two penalty attempts, centre McCabe was sinbinned after a try-saving tackle on Puma winger Gonzalo Camacho and number eight Radike Samo dropped the ball forward as he was trying to touch down.
The Australians were almost monopolising possession, however, and Barnes levelled the scores after 26 minutes only for sloppy defence from fly-half Quade Cooper to allow Hernandez another penalty to put the Argentines back in front three minutes later.
Cooper made some amends after 36 minutes with a superb short pass to send Sharpe charging for the line and the big lock reached his target only to be prevented from grounding the ball by Camacho.
The visitors went in 6-3 up at half-time and the first nine minutes of the second half followed the same pattern as the first, with the Wallabies enjoying plenty of the ball but making errors at key moments.
Barnes again levelled the scores at 6-6 after 49 minutes but the parity was to last only seconds.
Another casual clearance from Cooper was brilliantly charged down and collected by replacement loose forward Leonardi, who touched down for the first try of the match.
Two minutes later and winger Juan Imhoff returned a poor Barnes kick, racing through the midfield and down the sideline before flipping the ball inside for Farias Cabello to flop over the line.
Hernandez missed both conversions but made no mistake with his fourth penalty attempt and Argentina were 19-6 ahead with 24 minutes remaining.
McCabe, returning to the Wallaby midfield after injury, finally found a way through the Puma's defence in typically direct style to score Australia's first try after 60 minutes with Barnes adding the extras.
Phipps made the second try by bursting down the blindside before passing to Ioane, who cut inside the final defender to score. The conversion from Barnes put the home side ahead for the first time.
There were still 11 minutes to negotiate with a slender 20-19 lead but replacement full-back Kurtley Beale eased the home nerves with a penalty a minute from the hooter.