Captain Morgan hit 92 and Willey (12 not out) launched John Hastings for six to take the hosts to their target of 300 with 10 balls to spare, setting up a deciding match in Manchester on Sunday.
"Fortunately there wasn't much left to do when I batted," said left-arm paceman Willey. "The important thing was not to give my wicket away.
"After two losses, to level the series is a great set-up for Sunday. It should be exciting."
Glenn Maxwell was Australia's stand-out player and took an extraordinary catch late on.
He plundered 85 off 64 balls and shared a fourth-wicket partnership of 137 with George Bailey (75) after Willey had taken three early wickets to reduce them to 30 for three.
The touring side then slumped to 215 for seven before Matthew Wade (50 not out) and Hastings (34 not out) clubbed 84 in the last eight overs to lead Australia to a competitive total of 299 for seven.
Watched by British Prime Minister David Cameron, England lost Alex Hales early but Jason Roy (36), James Taylor (41) and Ben Stokes (41) helped Morgan keep up with the required run rate.
Morgan struck two sixes and eight fours and looked to be leading his side home until he was brilliantly caught one-handed by Maxwell at backward point off Pat Cummins.
Jonny Bairstow chipped in with 31 and Liam Plunkett 17 off 10 balls before he fell to another superb piece of fielding from Maxwell who knocked the ball up before falling over the boundary and took it one-handed as he leapt back on to the field.
Moeen Ali (21 not out) and Willey kept cool in mounting tension, the latter ending the match in spectacular style with a powerful strike over long-on to complete a triumphant return to the side.
"It's very disappointing," said Australia captain Steve Smith. "I thought 30 for three was tough to come back from but our middle order played well.
"Bailey and Maxwell's was a magnificent partnership. Then our bowling didn't get the ball in the right place often enough, the first and last ball went for four too many times."
(Reporting by Ed Osmond, editing by Pritha Sarkar)
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