Their second innings threatened to go the same way at 15-2.
The 126-run partnership between skipper Root and Denly was at the heart of England's fightback and Denly says the dressing room is full of optimism.
England left the field visibly broken after their horror show on day two of the third Ashes Test.
Earlier, Stokes had been at fault over the run-out of Jos Buttler, leaving England 253 for six and all but out of contention. "The facts are we've been in a position to win every Test match that we've played in so far".
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"I rate our chances very highly".
Speaking on Sky Sports after the game, Stokes called his stunning innings "unbelievable" and said it was "one I'll never forget".
"If we can knock them over in the morning, I've seen Test cricket have some unusual endings".
"I think if we start well again next innings, they might think "here we go again" so it's about creating that doubt in the mind, I think, through the way we've gone about it, is not letting the foot off, just keep going".
Graham Thorpe offered no excuses for England's abysmal batting after it was skittled out for 67 but stressed "these are the best players we have" with Australia looking certain to retain the Ashes.
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England had been 17 runs away from victory when Stokes offered a chance off the bowling of Cummins, with the ball swirling in the air as Marcus Harris sprinted in from the deep.
Left-hander Rory Burns (seven) edged Pat Cummins to David Warner at first slip before Surrey team-mate Jason Roy (eight), yet to convince as a Test opener for all his World Cup heroics, was bowled playing down the wrong line for by Josh Hazlewood, who took 5-30 in the first innings. That'll be what we're trying to do tomorrow, trying to make sure we're shutting down the scoreboard and making sure we're bowling balls in good areas with that new ball. According to ESPNcricinfo's figures, his percentage of balls left (35.1% of 416 balls faced in the series so far) ranks him third out of the top five leavers of the ball in a Test series over the past decade.
But when Hazlewood defied the increasingly slow pitch by getting a bouncer to lift, Denly was in no position to hook and gloved the ball to Australia captain and wicketkeeper Tim Paine, with England now 141-3. If these lapses sound frequent, Labuschagne had not given a chance in either the second innings at Lord's or the first at Headingley, showing that fortune not only favours the courageous, but also the prudent.
This was only the fourth time a team had made more than 300 to win in the fourth innings of a Test at Headingley.
He was cleared to bat on but, with just the tail for company, ran himself out when, deceived by a misfield, he failed to beat Denly´s throw to Bairstow.
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"I couldn't watch in the end I was just waiting to see what happened". But his arrival in England brought a refreshing change of scene, and the runs began to build up.