Needing a victory to stay in the series, England went to stumps on day two at 8-213, still 158 runs shy of Australia's first-innings 371.
On a day overshadowed by more DRS drama, Cummins took three wickets for the hosts and Lyon two as Ben Stokes' unbeaten 45 offered the only real resistance.
Thursday had loomed as a golden opportunity for England to take back some control, with temperatures exceeding 40C in Adelaide.
But after Mitchell Starc (54) helped Australia's tail wag for the first 40 minutes, England's batting again fell flat.
It was telling that Australia won the first two Tests without the injured Cummins, while Lyon bowled two overs in Perth and was dropped for Brisbane.
On Thursday, the pair made their presence known.
Where England were too short with the ball, Cummins (3-54) went full and nagged away outside off to draw the edges of Zak Crawley (9) and Joe Root (19).
The ball to dismiss Crawley was classic Cummins, angling in and doing enough off the seam to take the right-hander's edge.
And his removal of Root made it 12 times in his career he has dismissed England's best batter, having done so more than any other bowler in the world.
Lyon (2-51) took his wickets in the space of three balls in his first over. The double strike took his career total to 564 Test scalps and into second place on Australia's all-time bowling list. He is now one ahead of paceman Glenn McGrath and behind only legendary legspinner Shane Warne (708).
He first ensured Ollie Pope's miserable tour would continue, out for three playing a poor shot when he turned a ball straight to Josh Inglis at midwicket.
Lyon then came around the wicket to the left-handed Ben Duckett (29), drifted the ball into the England opener, then turned it away to take the top of off stump.
And after Cameron Green had Harry Brook caught behind for 45, it was Cummins who returned to get rid of Jamie Smith on 22 amid more DRS drama.
Australia's captain first thought he had his man caught at slip, before umpire Chris Gaffaney ruled the ball had come off Smith's helmet and not his glove.
There were questions over whether the catch carried, but the issue will raise further debate about Snicko given the audio spike appeared on the frame with the ball between glove and helmet.
Unlike Alex Carey's life on day one that cruelled England, where Snicko's operators admitted to human error, this one mattered little.
With the hosts employing a short-ball approach, Smith couldn't resist and a wild pull shot shortly after that went through to wicketkeeper Carey.
Again Snicko was called upon, but this time Smith was dismissed and England were faltering.
"That was extremely hot out there, it was hard," Lyon said.
"So I'm extremely proud of, especially the quick bowlers, the efforts that they put in.
"Consistently they put the ball in the right areas for long periods of time, back-to-basics type thing."
Carey, meanwhile, has had a superb series with both bat and gloves.
He had five catches on Thursday to go with his first-day hundred, with the fifth the pick of the lot as he went up to the stumps to Scott Boland to remove Will Jacks.
Boland also bowled Brydon Carse for a duck, swinging one back between the tailender's bat and pad.
And while Jofra Archer hit an unbeaten 30 late to support a cramping Stokes, Australia ended day two well on top and on track to keep the urn for a fifth straight series.
"We're behind the game again ... we're disappointed of course," England's batting coach Marcus Trescothick said.
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