A masterful unbeaten 89 from Morgan hauled England to 272 for five from their 50 overs after Australia captain Michael Clarke put the hosts in to bat under leaden skies.
Morgan’s effort helped England through a rain-affected innings, before the bowlers restricted Australia to 257 for nine in reply as Alastair Cook’s side claimed a convincing opening victory in the five-match series.
It was a frustrating day for Clarke’s top-ranked ODI side, who would be usurped by England if the hosts were to secure an unlikely 5-0 whitewash.
England have now won seven successive ODIs, and they have taken the last six home series in the format as the confidence within coach Andy Flower's ranks continues to grow.
After six successive centuries from England openers in ODIs, the run finally came to an end as Cook and Bell departed for 40 and 41 respectively.
It was a tricky opening period for England in dank, murky conditions as the players were twice forced off the field due to rain showers in a tough spell for the batsmen.
Jonathan Trott steadied the hosts with a typically assured 54 off 70 balls, and he anchored the innings as a pace attack of Brett Lee, Clint McKay, and Pat Cummins sought to exploit favourable conditions early on.
Ravi Bopara could only muster 15 before he was caught by Clarke off the bowling of McKay as England suffered a mid-innings blip, but Morgan proceeded to take centre stage.
The Middlesex batsman seized the initiative in the middle overs and consolidated the loss of wickets alongside Trott as the pair guided England into a strong position.
Morgan plundered five fours and four magnificent sixes as he took the game away from Australia with real poise and composure.
The unflustered batsman had the perfect foil in Trott, who was steady throughout his knock before he was finally bowled by spinner Xavier Doherty.
Wicketkeeper Craig Kieswetter joined Morgan, and the Somerset player provided a more aggressive approach as he pounded a quickfire 25.
All-rounder Shane Watson belatedly struck to remove Kieswetter with the final ball of the innings, by which time England had piled on the runs in a brutal onslaught led by Morgan.
Australia’s innings was opened by the destructive duo of Watson and David Warner, who quickly put England under the cosh with a series of lusty blows.
Watson, however, departed early for 12 as Finn had the burly batsman caught behind by Kieswetter.
Warner proceeded to unfurl a series of breathtaking blows with six fours and a maximum in his belligerent 56 with number three George Bailey (29) for company.
Bailey was the next to fall as James Anderson – who had to frequently leave the field due to illness – scattered his woodwork with a fearsome delivery.
Warner departed after reaching half-century as Anderson claimed another wicket with Kieswetter taking the catch behind the stumps.
Captain Clarke dropped anchor briefly as he consolidated with David Hussey, before the latter was bowled by the ebullient Finn.
Steven Smith and wicketkeeper Matthew Wade followed without providing a substantial impact – the latter after a horrendous mix-up with his captain as the pair looked to scamper through for an unlikely single.
Clarke was left dismayed as he was trapped flat-footed on the crease and dismissed plumb lbw by Bresnan for 61, and Australia’s innings proceeded to implode.
McKay only lasted six balls before he skied a short ball from Stuart Broad up into the air, while Pat Cummins was the ninth wicket to fall as he was trapped lbw by the Nottinghamshire paceman.
Lee was left unbeaten on 29 after a doughty display of aggressive strokeplay as England’s bowlers were left to take the plaudits for a distinguished all-round performance.
It was a chastening defeat for Australia in the end, and though the tourists were 15 runs short, the final overs played in a fairly subdued atmosphere due to the improbability of a successful run-chase.
England were buoyant as they celebrated an impressive victory at the start of the series, while Australia must now refocus ahead of the remaining four matches as they look to protect their number one ranking in the format.
The second one-day international is at The Oval on Sunday.