Archer took six for 62 and Curran marked his first appearance of the series with three wickets, but the prize scalp of Steve Smith evaded both men – Chris Woakes trapping him lbw for 80 deep into the second evening.
Remarkably that was his lowest score of a prolific series and Australia duly subsided 69 behind on 225 all out.
Smith had taken his series tally to 751 in six knocks, and his sequence of 50-plus scores against England to 10 in a row, but this time he was unable to fully make up for the mediocrity of his colleagues.
Denly, who became a father for the second time overnight, was badly dropped by Marcus Harris at slip and Burns was given lbw to the last ball, only for DRS to chalk it from the records.
The second six-for of Archer's England career was the result of some hostile and highly effective pace bowling, regularly delivered at over 90mph, and took his series haul to 22 at an average of 17.27.
Curran, meanwhile, fully justified his recall for a first Ashes cap, calling to mind his man-of-the-series performances against India last summer and raising question marks over his recent absence.
England began the day by adding 23 for their last two wickets, ending 294 all out. There were no further fireworks from Jos Buttler, who topped up his overnight score by six before being cleaned up for 70 by Pat Cummins.
Jack Leach played on next over allowing Mitch Marsh to celebrate a maiden five-wicket haul, earning the all-rounder the chance to lead his side off.
The turnaround brought David Warner to the crease, inviting England to heap more misery on the increasingly hapless opener. He arrived on the back of three consecutive ducks – marked by a trio of England fans in appropriate fancy dress – but at least managed to make five runs before being sent on his way.
For once it was not Stuart Broad who took him out, Archer finding the outside edge and Joe Root successfully overturning a not out verdict. There was initial scepticism over the UltraEdge technology, with suggestions an alternate noise had occurred at the precise moment ball passed bat, but it later transpired it was nothing more unusual than an old-fashioned nick.
There was no doubt over the contact when Archer did for Harris, a thick edge well picked up by Ben Stokes at second slip to bring forth Smith at 14 for two.
Archer greeted him with a fizzing 90mph bouncer and drew a careless waft second ball, before Curran's left-arm swing brought two lbw appeals.
Marnus Labuschagne edged Curran wide of Stokes on 15 but Australia's most obdurate pair went on to share a stand of 69 either side of lunch.
Archer finally parted them, first pinging Labuschagne on the elbow then capping a fiery afternoon spell by trapping him lbw for 48.
Smith, meanwhile, was only really troubled when Jonny Bairstow's arrival at the stumps led to him diving awkwardly to beat a throw that was instead collected at the non-striker's end.
Smiles exchanged with the England wicketkeeper, Smith dusted himself down and passed 50 by carving Leach for six.
Matthew Wade became Curran's first Ashes wicket before tea to leave Australia 147 for four, and from there the home side seized the initiative.
Marsh's batting could not match his impact with the ball, yielding just 17 before he turned the deserving Archer tamely to fine leg, while Curran bested Smith on 66. Flashing hard he sent a high chance towards Root at first slip but, possibly distracted by Stokes, the England skipper could not get a clean hand on it.
Undeterred, Curran settled for working away at the other end. Australia captain Tim Paine managed just one before feathering through to Bairstow and, feeding on the sudden rush of momentum, Curran made it two in two balls when a hooping yorker thumped Cummins in front.
The hat-trick ball was straight but overpitched, though the youngster had moved the dial.
England still needed Smith, though, and they got him in unexpectedly prosaic fashion when Woakes hit the knee-roll with a stump-to-stump offering.
Woakes was far from his best for the majority of the day, but one ball was more than enough to weigh against the remainder of his 10-over stint.
There were still two wickets to get and Archer made sure they were his, smashing Nathan Lyon's off stump before Burns claimed Peter Siddle with a stunning one-handed catch taking off at gully.