It was the heaviest defeat ever suffered by Australia - who remain number one in the 50-overs game - as England extended their unbeaten run to 10 one-day wins in a row.
Cook and Bopara kept their cool as they shepherded England to a Duckworth-Lewis target of 138 with seven wickets and 11 balls to spare in the rain-shortened match under the Old Trafford lights.
Man-of-the-match Bopara took two wickets for eight runs in four overs with his medium-pacers, before making 52 not out off 56 balls during his batting sting.
And afterwards the captain hailed his Essex teammate, saying: "We all know what a quality player he is, and he is starting to deliver for England now."
"The pitch was a little bit 'stroppy' but I think it got better as the night went on," said Bopara afterwards.
"It's about time I was there at the end!
"It was important that I got an early knock and that one at the Oval (82 from 85 balls in England's six-wicket win in the second match) gave me confidence and stood me in good stead for the series."
Before the match, Australia coach Mickey Arthur had called on his side to show some "mongrel" but there was precious little in evidence in this day-night fixture as the visitors endured a hapless collapse - including two run-outs.
Despite a series of plays and misses, from David Warner at Steven Finn in particular, no chances were given until Matthew Wade edged first-change Stuart Broad high and wide to a diving James Anderson at a solitary slip.
The wicketkeeper-batsman was put down on four, and one run later had another escape when this time an edge at Finn carried low to Tredwell but did not stick at second slip.
Warner clubbed a straight six off Broad, only to then have a let-off of his own on 30 when he upper-cut the same bowler high to third-man, where the ball slipped through Samit Patel's hands.
The fielder and England were soon able to breathe a sigh of relief, though, as the drop cost them just those two runs - before off-spinner Tredwell, in his first match since last year's World Cup, struck with his fourth delivery when he had Warner lbw sweeping.
For good measure, Patel and Tredwell then combined to run out number three Peter Forrest - trying to scurry a second for Wade to long-on but unable to beat the throw from the deep and neat work from the bowler to break the stumps.
Tredwell upped the ante with his next ball, as a frustrated Wade went on the charge and failed to get to an off-break and was stumped for a stodgy 12 from 40 balls.
Clarke pushed a faulty single to cover, and was easily short of his ground when Eoin Morgan produced a direct hit.
Then the introduction of Bopara ended an attempted fightback, as Steve Smith edged the medium-pace golden arm's first delivery down the leg-side to Craig Kieswetter, and the all-rounder struck with his first attempt in his second over too - David Hussey also caught-behind, trying to steer runs to third-man.
From 86 for six, Bailey and the tail tried to somehow salvage a defendable target - and with a six each off Tredwell and then Anderson from Australia's main contributor, they managed 58 for one wicket in the last 10 overs.
Australia captain Michael Clarke praised England but said his side would be stronger come their return for next year's Ashes Test series.
"We have no excuses, we came here to win all five games," said Clarke.
"I don't want to take anything away from England, but I will make sure we are as well prepared as we can possibly be for the Ashes."
Follow us @SprtNationalUAE