Responding to Australia's formidable declaration mark of 497 for eight, England closed on 200 for five, still 98 short of the follow-on target at Old Trafford.
Their hopes of saving the game and the urn were aided by conditions in Manchester, with morning rain and bad light in the evening allowing just 64 overs possible out a planned 98, while for long periods Burns (81) and Root (71) appeared in control of matters.
In the end the pair shared a defiant stand of 141 in nearly 53 overs – their second-century partnership and England's biggest of the series – only for Hazlewood to intervene with a late, decisive spell.
Having already seen off nightwatchman Craig Overton in his first over of the day the seamer returned in the final hour to wreck England's good work.
First he had Burns caught in the slips, his only authentic chance in 185 balls at the crease, then roared in to trap Root lbw in his very next over.
He had one more trick up his sleeve before the umpires led the sides off, nipping one in off the pitch to tear out Jason Roy's middle stump and complete a drastic turnaround.
For England the main source of solace is Ben Stokes' presence at the crease but the hero of Headingley will need to turn in yet another brilliant performance to get his team out of jail here.
Heavy showers wiped out the morning, with play not beginning until 1.30pm. After several hours of pondering and planning how he might frustrate the bowlers, Overton proceeded to nick the ninth ball of the day.
It was Hazlewood's first intervention on his way to figures of four for 48, a performance that might yet have settled the direction of the match.
Steve Smith, fresh from his latest epic innings of 211, was on hand to hang on at second slip for his 100th Test catch.
And so began a profitable, and highly accomplished, stand between two men who would go on to see of 353 deliveries between them. It was not always comfortable, both men wearing their share of bruises and Root even banking a broken box after one unpleasant blow from Mitchell Starc, but it was exactly what their country needed.
Hazlewood tested Root early on, beating the bat and seaming one perilously close to the off stump when he offered no shot, while Burns quickly got used to a steady diet of bouncers as Australia sought to challenge his perceived weakness.
But the bowling got as they chased another breakthrough, both batsmen offered cheap runs on the leg side as the boundary count began to mount.
By the time they had spent 20 overs together the partnership was worth 50 and Starc's second spell of the day hastened things along nicely, shipping 22 runs from two overs as Burns and Root began to enjoy themselves.
By the end of an elongated afternoon session they had added exactly 100.
There was a growing sense of enjoyment in the stands, with Nathan Lyon picked out as the butt of a persistent joke as fans cheered his every take at the bowler's end. He has surely not forgotten his botched run-out in Leeds, but neither have the English public.
It took a wonderful spell from Pat Cummins to recalibrate things, Root nicking clean between wicketkeeper and first slip on 54 – neither of whom moved a muscle – the first genuine chance of the stand.
Next up he pounded Root on the knee-roll – spawning a misguided review from Paine, his 10th such failure of the series but also some serious discomfort for the batsman – and rapped both batsmen on the glove.
He had ramped up the pressure and Hazlewood cashed in. Just before 6pm he persuaded Burns to nick off for the first time in a measured innings that confirmed his suitability for the highest stage, with Smith making sure once again.
The burden now lay weighed heavily on Root but he was beaten by a rapid delivery that jagged into his pad and left him plumb lbw in Hazlewood's next over.
The pair had batted with caution and class to suck the momentum away from Australia but in a few minutes the foundations had crumbled.
Roy, moved down to the middle order to unlock his run-scoring potential, hit three boundaries as he zipped along to 22 but was found wanting. Having just crashed Hazlewood to the ropes he was ruthlessly unpicked, bent double by a seaming delivery that cleaned him up.
That left Stokes and Jonny Bairstow in the middle, with a monstrous task awaiting.