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The Australians amassed a massive 303-run innings lead in blistering heat and then reduced the battle-weary tourists to 93 for four at the close with a day to play.
Root was unbeaten on 42 at the close with Jonny Bairstow not out 17.
"I think Gaz (Lyon) is the key tomorrow. The wicket is definitely suiting his conditions, especially with the left handers, he'll come into play," Australian centurion Shaun Marsh said.
"I thought he bowled really well today and showed his class and hopefully he can come out tomorrow and get a few early breakthroughs for us."
Australia called a halt to their epic first innings at 649 for seven, and soon had England in trouble after the tourists had been cooked in the field for more than two draining days.
But Cook's reprieve did not last long and he was bowled for 10 when a sharp Lyon turner hit the top of off-stump in the off-spinner's first over.
Cook, who became during his innings only the sixth batsman to score 12,000 Test runs, finished the series with 376 runs at 47.
Dawid Malan followed when Lyon trapped him leg before wicket for five.
- Ton-up brothers -
"In many ways the last few days have summed up our trip," said England assistant coach Paul Farbrace.
"It's been exceptionally tough, we've come up against a team that's playing very good cricket but the one thing that we've been talking about is making sure we keep on fighting and battling."
Shaun and Mitchell Marsh earlier became the third set of brothers to score centuries in the same innings for Australia.
The Chappells were the first Australian brothers to notch centuries in the same innings at The Oval in 1972 with the Waughs the last to do it against England, also at The Oval, in 2001.
Mitchell Marsh almost spoiled his century in all the excitement as he embraced his brother mid-pitch before he had completed the second run scrambling back to his crease to avoid being run out.
He was the third centurion of the massive Australian first innings total, following Usman Khawaja's 171 on Saturday.
Shaun Marsh was eventually dismissed after lunch, run out for 156 by Stoneman's direct throw from the covers.
He batted for 403 minutes and faced 291 balls for his second century of the series and his sixth in 28 Tests.
Starc slogged Moeen Ali for six and was out the next ball finding Vince at long on for 11.
Big-hearted paceman James Anderson bowled with great economy in the stifling conditions as temperatures hit 42 Celsius (108 Farenheit), taking one for 56 off 34 overs.
Spinners Ali and debutant Mason Crane conceded a combined 363 runs in taking three wickets between them.