"They've got a very good bowling attack and it's one of those where I'm just delighted to reach three figures for England again."
Bairstow celebrated his seventh test century by running with arms aloft halfway toward the England team's dressing room, where his teammates had gathered to acknowledge a vital innings in the context of the match which had earlier looked to be slipping away. England has already lost the five-match series with three straight losses, allowing Australia to retain the Ashes, but are determined to get something positive out of the tour.
At stumps, Bairstow was 103 not out and Jack Leach was on four, with England trailing Australia by 158 runs.
Earlier, Stokes made a typically swashbuckling 66 of 91 balls, including nine fours and a towering six over cover, as part of a 128-run partnership with Bairstow which steadied England's innings after its early collapse.
"The partnership with Ben was a big one and it was tough there and I'm delighted with it," Bairstow said.
Nathan Lyon eventually ended the 128-run partnership by trapping Stokes lbw with a ball that kept low.
He was perhaps fortunate to get that far having before tea survived a missed caught-and-bowled chance off Cummins and then successfully reviewing an on-field lbw 'out' decision where the ball had in fact missed the pad and flicked the off stump but not dislodged the bail.
Shortly after raising his half-century off 80 balls, Bairstow was struck by a ball from Cummins that reared up off the pitch and struck the batsman on the thumb. Bairstow received treatment on the ground before resuming his innings, but was visibly in discomfort holding the bat.
"It takes quite a bit to get me off the park," Bairstow said. "Yes it was sore but in the circumstances of the game it was a decision I made to stay out there.
"You're out there playing in a New Year's Ashes test match in Sydney in front of a big crowd and it's going to take a lot to take you away from that."
Jos Buttler became the second duck of the innings, chipping to Khawaja at cover off Cummins (2-68) to reduce England to 173-6 and in danger of not avoiding the follow-on target of 218.
But Bairstow and Mark Wood combined for a 72-run partnership to reduce the visitors nerves. Cummins eventually had Wood caught by Lyon for 39 off 41 balls going for one big shot too many.
Before lunch, England's batting frailties again were exposed by Australia's pace attack after it resumed at 13 without loss in a rain delayed session.
Haseeb Hameed was dropped on two by wicketkeeper Alex Carey in Mitchell Starc's second over, but the reprieve only cost four runs as Starc (1-49) bowled the opener for six in his next over.
Scott Boland (2-25) bowled Zak Crawley (18) and then had Root caught at slip for a duck in his next over with the England skipper again playing away from his body and edging behind to Steve Smith at second slip.
Green dismissed Dawid Malan (3) in the over before lunch, leaving England in tatters at the break.
"The first session was absolutely brilliant bowling," Australia coach Justin Langer said. "England had the better of us for the second two sessions.
"I thought Ben Stokes was courageous with his side (injury) and Jonny's Bairstow's effort after he copped one on the thumb was really courageous."
Boland was taken for a precautionary scan on his ribs after tumbling in his follow through with the final ball before tea. Australia team management later said the 32-year-old, who made a sensational debut in Melbourne to help Australia retain the Ashes, had "been cleared of any damage" and later returned to the match.