A stirring fightback wasn't enough on Sunday at Tokyo Stadium, as Australia battled back in vain from 18 points down soon after halftime, closing to within a point entering the final 10 minutes.
They couldn't score again, an outcome that means they'll probably finish second in group D and, if other results fall as expected, face England in the quarter-finals and potentially New Zealand in the semis.
The Wallabies scored three tries to two and were the vastly superior team in the second half, but they paid for an error-riddled opening 40 minutes which lacked purpose and execution.
Their other bugbear was long delays that halted momentum, when French referee Romain Poite and Kiwi TMO Ben Skeen ruminated on a number of collision incidents.
Most rulings went against Cheika's men, most notably when centre Samu Kerevi was penalised for raising his forearm to the chest of replacement Welsh five-eighth Rhys Patchell.
It proved match-swinging as Patchell landed the subsequent penalty and a rejuvenated Wales scored a try in a key period just before halftime.
"As a former rugby player, I'm embarrassed about that," Cheika said.
"You've got to look after players but not to an extreme when you're looking after players for doctors and lawyers.
"They (match officials) all seem spooked, like they're all worried about things so much. I'm not sure why they're worried. Players aren't worried.
"They're making decisions on all types of crazy stuff."
Cheika has other issues to address after re-introduced halves Will Genia and Bernard Foley failed to stamp their authority.
It mirrored last week's struggles for starting pair Nic White and Christian Lealiifano in the win over Fiji.
Matt Toomua's injection off the bench soon after halftime provided a playmaking spark. He might have forced Cheika's hand for next week's match against Uruguay.
Wales opened the scoring inside 35 seconds through a Dan Biggar field goal.
Both teams notched their first tries through cross-kicks.
Hadleigh Parkes out-jumped Marike Koroibete to put Wales 10-0 clear before Adam Ashley Cooper latched onto a clever Foley kick in the 20th minute.
A Foley penalty goal drew Australia to within two points before Patchell slotted two himself and halfback Gareth Davies bagged a crucial try on the stroke of halftime to put his team 23-8 clear.
Davies raced 60m after timing his run perfectly to intercept a Genia pass.
A Patchell field goal pushed Wales further ahead before Toomua sparked the try of the game, making two breaks in the lead-up to the impressive Dane Haylett-Petty crossing.
Australia's mortgage on possession and territory in the second spell set up a burrowing try to captain Michael Hooper to reduce the deficit to four points.
Wales halted the momentum through a Patchell penalty and defended fiercely in the final minutes.
Patchell was a key figure, having entered on the half-hour mark after Biggar suffered a concussion when pulling off a try-saving tackle on Kerevi.
The defeat is just the third Australia have suffered in any World Cup pool phase, after losses in 1995 to South Africa and 2011 against Ireland.