In-form off-spinner Nathan Lyon won a successful leg before wicket review to remove Cook (16), who shared in a 53- run opening stand with Stoneman.
Paceman Mitchell Starc then had Stoneman snapped up in the gully for 36 to leave England rocking at 54 for two shortly before the dinner break.
At dinner, England were 68 for two with skipper Joe Root shouldering a heavy responsibility on seven and James Vince not out eight ahead of the pivotal final twilight session under lights.
The Australians were knocked over for 138 off 58 overs at tea and after leading by 215 runs on first innings pushed their lead past historical proportions at the ground.
The highest winning fourth innings at Adelaide Oval stands at 315 for six by Australia against England in 1902. England's highest run chase at the ground is 109 made in 1912.
Cook survived a confident lbw appeal off Josh Hazlewood when he was on one, but the Australians on that occasion did not seek a review and replays showed the ball hitting leg stump.
Cook was Lyon's 56th wicket this calendar year making him the leading wicket-taker in Test cricket in 2017.
England's all-time leading wicket-taker James Anderson took five for 43 -- his first five-wicket haul in Australia -- while fellow paceman Chris Woakes captured four for 36.
Usman Khawaja and Starc were joint top scorers with just 20 each in Australia's modest second innings.
England kept the Australians under pressure after they resumed the day at 53 for four.
Nightwatchman Lyon hit a couple of fours and was struck on the helmet grille by Stuart Broad before he holed out to mid-off off Anderson for 14 for Tuesday's first wicket.
Peter Handscomb, the other overnight batsman, struggled before he aimed to steer through the slips cordon only to be caught by Dawid Malan for 12 in 76 minutes and give Anderson his fourth wicket.
Tim Paine, who scored a half-century in the first innings, was dismissed for 11 top-edging a pull shot off Woakes and Craig Overton took a splendid sprawling catch running in from fine leg.
Starc launched a six over midwicket off spinner Moeen Ali before first innings centurion Marsh was cleaned up by Woakes for 19, ending a 32-run partnership with Starc.
Anderson made a valiant effort to catch Starc on 18, haring down the pitch off his own bowling but he could not hold a difficult two-handed chance as he slid into the stumps, tearing his trousers.
But Anderson got his fifth wicket of the innings when Starc holed out to Moeen for 20.
Hazlewood was the last wicket to fall, caught in the gully off Overton for three, leaving Pat Cummins not out 11.
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