Local boy Andrew Tye had positioned himself as a fitting hometown hero, taking five wickets as the tourists were dismissed for 259, but Curran saw his effort and trumped it with a nerveless 5-35 to seal a 12-run win for the tourists.
Eoin Morgan’s side had Joe Root’s industrious 62 to thank for mustering what was little more than a passable total, with a 4-1 series win partially avenging their Ashes defeat.
Curran’s exuberant celebrations - arms flung wide to mimic an aeroplane - were richly earned as he claimed four of the last five wickets in a tense finale, closing the campaign in style by bowling Tim Paine.
Steve Smith had put England in to bat for the second time in three days, doubtless hoping to recreate some of the mayhem from Adelaide, where England lost their first five wickets for just eight runs.
Jason Roy ensured there would be no repeat of that horror show, taking a rare all-run four off the first ball at the new ground then flaying half-a-dozen boundaries and one muscular six off Mitchell Starc.
Roy had one major slice of luck when he was caught behind from a no-ball but looked set fair until he aimed a booming drive at Tye and skied a catch to mid-off.
The baton passed to Jonny Bairstow (44) and he had just hit fifth gear - crunching Glenn Maxwell for six then uppercutting Starc for another - when an inside edge off pegged back leg-stump.
England never fully regained their poise but Root played a typically industrious anchoring role, picking gaps and dashing hard for his runs.
Alex Hales (35) was hurried by a Mitchell Marsh bouncer, Morgan drew grimaces with an ugly shot on the charge and Jos Buttler holed out to Tye moments after a punishing blow down the ground.
After Moeen Ali and Adil Rashid also failed, it was left to Tye to take the stage.
He finished things with three wickets in seven balls, deceiving David Willey with a change in pace, finally persuading Root to take a risk too far and castling Jake Ball with a triumphant yorker.
Curran’s dream day should have started with his fourth delivery, Bairstow fumbling a catch at slip, but he still scored an early success by cleaning up the off-colour David Warner.
The free-hitting Marcus Stoinis eased the pressure with a few clean strikes and had Morgan not run out Head with a direct hit, the game could have drifted quickly.
As it was England began to probe the seam of vulnerability that runs through Australia’s one-day batting.
Smith, no longer the dominant figure he was during the Ashes, was stumped off Rashid and Marsh fell victim to a staggering return catch by Moeen.
The umpire had already taken evasive action to avoid being floored by blistering drive when Moeen threw himself backwards and flung a hand out to grasp the ball triumphantly in his fingertips.
Stoinis (87) refused to let that moment be the turning point, adding 56 with Maxwell but his demise - bunting Rashid’s full-toss straight down Curran’s throat in the 34th over - ushered in further angst.
Curran was the architect, trapping Maxwell in front and taking Starc’s edge in the space of three balls. When Moeen picked off Tye, Australia required 53 with two wickets in hand.
Moeen thought he had one of them but Hales reprieved Paine on eight, stooping low at long-on. The match ended with a tone of semi-organised chaos, England adding further misfields, Adam Zampa overturning an lbw decision on DRS and Ball completing his spell while afflicted by a sudden bout of illness.
Curran provided the calm that others lacked, bowling Zampa and then Paine to carry his team across the line.