Morgan struck 89 off 63 balls to lift the hosts to a total of 272 for five and the touring side always threatened to reach their target thanks to fluent half-centuries from David Warner and captain Michael Clarke [ Images ].
The players were forced off three times due to rain but Cook made 40 and Bell 41 to give their team a good platform.
Jonathan Trott contributed a useful 54 off 70 balls before the fireworks were provided by Morgan.
The diminutive left-hander struck four sixes and five fours in a fierce late assault on the Australian bowlers to lift England to an imposing total.
Australia lost the wicket of Shane Watson [ Images ] early in their chase but Warner made 56 off 61 balls to give them momentum.
However, he edged a catch to wicketkeeper Craig Kieswetter off James Anderson [ Images ] and George Bailey, David Hussey [ Images ] and Steve Smith fell cheaply to leave the Australians struggling on 147 for five.
Clarke then found a useful ally in Matt Wade and the pair added 57 for the sixth wicket to give Australia hope.
Wade was run out for 27 when he failed to respond to Clarke's call.
Brett Lee [ Images ] made a defiant unbeaten 29 but Clint McKay and Pat Cummins fell cheaply and England's experienced bowling attack maintained control in the closing overs to seal victory.
The second game in the five-match series is at The Oval on Sunday.
Photo: England's Eoin Morgan hits out during the first one-day international against Australia at Lord's cricket ground in London [ Images ] (REUTERS/Philip Brown)