Pacer Wood took 5-34 and 2-66 while Woakes had a six-wicket match haul that included key batters Marnus Labuschagne in the first and Usman Khawaja in the second.
Australia leads 2-1 with the fourth test at Old Trafford starting on July 19.
By lunch, England had moved to 153-4 and within 98 runs of its target, four years on from a famous Stokes-inspired victory at the same venue.
But the match swung Australia’s way early after lunch when Stokes was caught behind for only 13 and Jonny Bairstow (5) was bowled — both against Starc, who finished with 5-78.
Stokes’ dismissal was a soft nick down the leg-side robbing the hosts of their inspirational captain and the architect of their 2019 Ashes miracle.
Earlier, England’s openers resumed in cautious mood at 27 without loss, adding 15 careful runs to the score before Ben Duckett (23) was trapped in the crease by Starc, lbw to a ball destined for leg stump.
Moeen Ali was the man to replace Duckett at No. 3, a surprise switch with Brook and a hefty promotion from No. 7. The all-rounder has done the job before, but last appeared this high in November 2018.
It was a typically bold gambit from the Brendon McCullum regime but one that did not pay off for Ali though it did allow Brook to bat at No. 5. Ali made five from 15 deliveries and had no answer for a 90 mph rocket from Starc, forcing its way through a modest drive and pounding the stumps.
Brook’s stand of 38 with Root was beginning to give the chase a sense of security, but the latter was undone unexpectedly with the lunch break in sight. Swiveling into a pull as the returning Pat Cummins dragged one down, Root gloved it through to the keeper for a tame ending on 21.
Zak Crawley’s growing confidence cost him his wicket on 44, throwing himself into a drive off Mitch Marsh and feeding the edge through to Alex Carey. At 93-3, both teams were moving toward their destinations.
England has chased down four bigger totals in the “Bazball” era, including 296 at the same ground last year, but in keeping their hopes of reclaiming the urn afloat this is likely its most satisfying pursuit yet. After squandering promising positions at Edgbaston and Lord’s, England are finally on the board and will take renewed belief into the remaining clashes at Old Trafford and the Oval.
“It seems like a series where each session swings in one team’s favour and then the next session the other team picks it up,” Cummins said. “I was just chatting to Stokesy - I wouldn’t mind a stress-free one. We’ll take a few days off and go and recharge the batteries. People are flying off to various places and we’ll come together all excited for Manchester.”
The last 12 runs came in a frenzy, short balls flicked and pulled in all directions and Carey — whose controversial stumping of Bairstow at Lord’s added an edge in the run-up to this test — delighting the crowd with a late drop. But the final moment was all class, Woakes flaying Starc for four in a rousing win and the latest twist in a superb Ashes series that deservedly stayed alive.
