The two sides went into Sunday's (AEDT) encounter knowing victory would see them avoid South Africa in the quarter-finals and set up a last-eight clash with Scotland.
Australia had the better of a tightly contested first half but appeared set for defeat when they saw scrum-half Will Genia and lock Dean Mumm sin-binned in the space of three minutes after the interval.
Despite being reduced to 13 men, Australia stood firm against incredible pressure on their line from Wales, denying Wales beyond the tryline on three occasions.
"It's pretty hard to make comments after that because maybe it wasn't absolutely perfect in the way we played the game and we proably made some mistakes and have got so much improvement (in us) but for our courage and the way we put our bodies on the line and defended, I was very proud of the lads," Cheika said after the match.
"I think our finishers did really well, they came on and sured up things, during that period Rob Simmons, James Slipper, Matt Toomua, Kurtley Beale, Tatafu Polota-Nau and Greg Holmes all came on in a really difficult situation and worked really hard.
"And that's their job and I'm not quite sure how we managed it because Wales played very well and full credit to them because they took it to us."
Ben McCalman's try-saving tackle on George North was immense and Bernard Foley's five penalty goals sealed victory in a thrilling encounter and ensured Australia a return to Twickenham to face Scotland next Monday (AEDT).
Wales now face the daunting task of trying to seal a second straight win over South Africa after beating them in November last year having previously lost their last 16 matches against the Springboks.
Dan Biggar slotted over a fourth-minute penalty, but Australia eventually began to dictate matters in the breakdown and in the scrum and levelled matters in the 24th minute thanks to a Foley kick.
Further pressure from Cheika's men allowed them to take the lead through Foley, but their advantage did not last long as Biggar nailed a three-pointer from long range three minutes later.
Again Wales' indiscipline inside their own half proved costly as Foley restored Australia's advantage with a penalty from out wide following foul play from Taulupe Faletau.
Biggar missed for the for the first time in the tournament a minute before half-time, but Australia failed to capitalise, Matt Giteau's effort from 57 metres falling short.
Australia remained in the ascendancy in the second half and Foley gave the Wallabies breathing room with a simple penalty due another indiscretion from Faletau, who was penalised for a neck roll.
However, the game appeared to turn in Wales' favour as Genia first saw yellow for tripping and Mumm each saw yellow.
First Genia was sent to sin bin for preventing Gareth Davies - who had earlier charged down a clearing Foley kick to give Wales territory - from taking a quick-tap penalty.
Faletau was then ruled by the TMO to have fumbled the ball over the line before Mumm followed Genia to the bin for an infringement in the line-out.
Sensational defence from Australia stopped George North from finally capitalising on Wales' extra men following quick hands from the backline, and Jamie Roberts was too held up as the Wallabies showcased remarkable resolve.
When restored to 14 men, an excellent Adam Ashley-Cooper tackle as Wales looked to spread it wide once more allowed the Wallabies to win a penalty at the breakdown and clear for territory.
Foley slotted over his fifth penalty from in front to give the Wallabies an decisive nine-point lead, with his late miss after Alex Cuthbert saw yellow for a deliberate knock-on proving immaterial.