The Pumas head back to Argentina for their final two matches still searching for their first win of the Rugby Championship with only one victory, over Georgia, in seven matches this year.
Cheika was at a loss to explain why his side were so insipid in the first half, after which they trailed 13-10, before catching fire in the second.
The Wallabies have been beaten twice by New Zealand and once by South Africa, but they eased past the Pumas thanks to six tries in Canberra.
Full-back Israel Folau scored a double try with Australia's Sekope Kepu, Will Genia, Nick Phipps and Jordan Uelese recording the other tries.
Foley had a big game, kicking at 100 per cent to contribute 15 points and also pulled off a try-saving tackle in the second half.
"At halftime I didn't have to say anything", Cheika told Fox Sports' Rod Kafer after the match.
"I'm not sure why because it's a Test match, you're playing for Australia but they sorted themselves in the second half and that was better rugby from us".
"You have to set the platform or you get nothing out wide, and then you get some straight hard running, better defence to put pressure on the opposition and you get a better outcome".
The Wallabies finally broke through in the Southern hemisphere tournament after some frustrating close shaves.
But he felt a lift in their intensity and urgency in the second half was key to the Wallabies revival, with the victory coming after last week's disappointing draw with South Africa.
Argentina replied with tries by scrumhalf Martin Landajo and wing Matias Moroni, while flyhalf Nicolas Sanchez was on target with both conversions and two penalties.
Cheika wasn't particularly happy with the first-half efforts of his back-row of skipper Michael Hooper and Ned Hanigan and No.8 Sean McMahon, who were outplayed by the Pumas.
The home side hit back just five minutes later as the ball was sent out to the right to Folau, who went over with Foley tying things up with the conversion.
The Wallabies delivered a rousing second half against Argentina to secure their first win of the Rugby Championship on Saturday.