Morgan (103no) and Luke Wright joined forces at 97 for four in pursuit of 267 for seven but, like Australia's Michael Clarke (87no) before them, refused to panic under pressure.
Clarke had shared half-century stands with Mike Hussey and James Hopes to ensure a testing target under lights. After the hosts then stumbled from 75 for one, their challenge had a still more significant look to it.
But a 95-run partnership in 15 overs kept them in with a near even chance – and after losing Wright, Morgan's uncanny ability to pace an innings and pick and execute the right shots helped England home with a misleading four overs to spare.
The Irishman's deft deflections and powerful drives brought him 16 fours from only 85 balls in his third one-day international hundred, and prevented his adopted country losing a ninth match in their last ten attempts in this format against Australia.
He still needed support from the lower order, though – and it duly came from Tim Bresnan, whose survival on 10 by a hair's breadth third-umpire run-out decision early in a match-clinching stand of 71 was an apt indicator of a closely-fought series opener.
England captain Andrew Strauss was full of praise for the Dublin-born 23-year-old Morgan, who last month helped England beat Australia to win the World Twenty20 tournament in the Caribbean.
Strauss said: "What a fantastic innings from Eoin Morgan – that was one of the best I've seen. Halfway through our innings it looked like we had some hard work to do. I think we always felt that 267 was very chaseable on that wicket."
Morgan was named man of the match for his display and was thrilled with his performance and an England victory. He said: "I've been in good nick for a while now and I want to play more cricket to take advantage of it. I think I try to keep things quite simple and it's easy to do when you're in good nick."
Australia captain Ricky Ponting felt his side's total was below par. "That sort of total was probably a little bit shy, it was a great wicket," he said. "The way he (Morgan] played tonight they deserved to win the game."
The second game of five, another day-nighter, takes place tomorrow in Cardiff.