Folau replaces Jesse Mogg, who has been benched, at the back with Nick Cummins moving onto the wing.
McKenzie has made four changes to his starting XV with Sekope Kepu coming in at tighthead prop replacing Ben Alexander.
Reds back-rower Jack Schatz is in line to make his Wallabies debut from the bench.
Cooper will reunited with his Reds teammate and good friend Will Genia at the scrumbase for the Wallabies who are desperate for a win after two successive losses to New Zealand to begin their Rugby Championship campaign.
McKenzie is confident the duo can ignite the Wallabies attack.
"We have very specific thoughts about how we want to play this game and feel Quade is best suited to carrying that out,'' Ewen McKenzie said.
"He's been part of our matchday squad in both games so far but his role now changes from being a finisher to starting the game. We feel that role will suit how we want to play.
"He is also an experienced and successful Test player, which is handy to have against a hardened Springboks outfit.
"It has been almost 12 months since he last started a Test match and I'm sure there were times where he wondered if he would wear the number 10 jersey again. He's worked really hard both on and off the field to now get that chance.
"He's carried himself extremely well all year and during the last fortnight in the role we've asked him to complete. Now, he gets a chance to make an impact from the start of the game.''
The Springboks opened their Rugby Championship campaign with successive wins over Argentina.
Australia: Israel Folau, Nick Cummins, Adam Ashley-Cooper, Christian Leali'ifano, James O'Connor, Quade Cooper, Will Genia. James Slipper, Stephen Moore, Sekope Kepu, Rob Simmons, James Horwill (captain), Scott Fardy, Michael Hooper, Ben Mowen. Reserves: Saia Fainga'a, Scott Sio, Ben Alexander, Kane Douglas, Jake Schatz, Nic White, Matt Toomua, Jesse Mogg.
-AAP