Cooper has been treated mercilessly by New Zealand crowds in the past for perceived wrongdoings against former skipper Richie McCaw, and even All Blacks coach Steve Hansen, who predicted a start for Cooper, called him a "maligned"player.
"Quade's a good player," Hansen said. "He gets maligned a bit over here because of a few incidents with Richie and we're not too forgiving of people who pick on Rich - I'm talking about fans - but within the team he's well respected. He's a good player, so he'll enhance them I think."
Cheika said Cooper, who now plays for French club Toulon, deserved his chance.
"Since he's been back he's been proactive in putting himself out there ... looking to better himself as a player," Cheika said. "Mate, I like him, I back him, I think he's a good person and a very, very good rugby player. Here's an opportunity to show that to as many people who are watching."
Cooper's in-form opposite Beauden Barrett said: "We know what he brings - X-factor - he's one to keep any eye on."
Asked if he hoped the Wellington crowd would treat Cooper with respect, he said: "I hope so. He's a good bloke and he deserves to be treated like anyone else. I know as a team we respect him."
Hansen is predicting Cooper and Foley to lead a two-pronged kicking attack at Westpac Stadium which is one of the reasons for the selection of Ben Smith at fullback.
The Wallabies were beaten all over the park last week, but they were hurt in particular by their inability to gain any territory and put the All Blacks under any sort of pressure. Instead of playing on the front foot, they were reduced to flinging passes behind the advantage line, presenting easy targets for a fired-up defence.
Smith's move to the back means Israel Dagg will start on the wing for only the third time in 53 tests.
"Both of them are world class fullbacks and we think Ben is certainly a world class winger," Hansen said. "We think we need to see if Izzie can start a test there and do a job for us because it opens up further down the track other selection possibilities. The way we play all our back three are fullbacks anyway, it's like playing three fullbacks.
"It may well also allow Ben to come into the game a wee bit more if Australia are going to kick which I think they probably will do a bit more."
Other changes for Australia see Samu Kerevi starting at centre and a return for loose forward Scott Fardy as Cheika searches for a first test win since their World Cup semifinal victory over Argentina.
Wallabies team to face the All Blacks in Wellington on Saturday night:
Israel Folau, Adam Ashley-Cooper, Samu Kerevi, Bernard Foley, Dane Haylett-Petty, Quade Cooper, Will Genia, David Pocock, Michael Hooper, Scott Fardy, Adam Coleman, Kane Douglas, Sekope Kepu, Stephen Moore (capt), Scott Sio.
Reserves: Tatafu Polota-Nau, James Slipper, Allan Ala'alatoa, Dean Mumm, Will Skelton, Tevita Kuridrani, Reece Hodge, Nick Phipps.
