The England skipper hit a run-a-ball 92 at Headingley, with James Taylor and Ben Stokes both scoring 41.
David Willey had earlier reduced Australia to 30-3 with the new ball before they recovered to post 299-7.
Glenn Maxwell, who was dropped on 6 and 35, top-scored with 85 off 64 balls.
The Australia all-rounder also took two stunning catches but his efforts were not enough to prevent England overhauling a score of 300 or more for only the fourth time in their one-day international history.
Their previous best ODI run-chase against Australia was 270-4 at Lord's in 1997.
It was England's second successive victory, after they also won at Old Trafford on Tuesday, and means the hosts can still send Australia home defeated across all formats after they won the Ashes 3-2 and the solitary Twenty20 international by five runs.
Morgan anchored England's run-chase perfectly with his eighth score of 50 or more in his last nine innings.
Jason Roy and Taylor had shared a first-wicket partnership of 72 off 53 balls but England's progress stalled when both were dismissed in quick succession.
Morgan, however, slowly rebuilt the innings as he took 40 balls to score his first 20 runs before slowly accelerating.
He hit John Hastings for a straight six which landed on the roof of the adjacent rugby stadium as he thumped eight fours and two maximums before being caught at backward point by Maxwell's stunning one-handed catch.
Although Morgan appeared disappointed not to have finished the job, his brilliant innings took England to 238-5 with more than 10 overs remaining.
Some big hitting from Liam Plunkett and Willey, plus a steady hand from Moeen Ali, saw England home with 10 balls to spare.
Morgan is the highest scorer in the series but only just ahead of Taylor, who followed up his century at Old Trafford with a counter-attacking 41 off 42 balls.
He hit eight boundaries in his innings - compared to just five in his career-best 101 in Manchester earlier this week - and displayed a different side to his game before chasing a ball down the leg side and being acrobatically caught by Matthew Wade.
Taking on the fast bowlers, he was particularly strong through the off side as he cut and drove impressively to seize the early initiative back from Australia following the early loss of Alex Hales for a two-ball duck.
Former England captain Michael Vaughan believes Taylor has now done enough to justify selection on the tour of the UAE to play Pakistan next month.
"I think they've got to take him," said Vaughan. "Taylor plays spin differently to many England players, using his feet well.
"He hasn't faced brilliant spin in this series, but he's played it in a good fashion, and he's looked good against the quick bowlers. He's read situations well, and that's what he'd have to do at five in Test cricket."
Willey, who will play for Yorkshire next season after deciding to leave Northants this summer, was superb with the new ball in the absence of the injured Chris Woakes and the rested Steven Finn.
The left-armer started by swinging the white ball into the right-handers' pads - dismissing Joe Burns (played on) and Steve Smith (lbw) cheaply - before occasionally running it across the batsmen.
That created doubt in the minds of Australia's batsmen and Aaron Finch was the next to go when he edged behind to Jonny Bairstow.
Willey and Moeen were the pick of the England bowlers, with Australia failing to score a run off 50% of the latter's deliveries.
However, the hosts' familiar problems at the back end of an innings resurfaced, with Australia smashing 77 off the last six overs.
Both Willey and his fellow opening bowler Mark Wood bowled too full, allowing Wade and Hastings to get under the ball and plunder enough boundaries to take Australia from 215-7 in the 42nd over to 299-7 when their innings closed.
It was not just Willey who impressed at Headingley, with another honorary Yorkshireman dazzling the crowd with some fantastic cricket.
Maxwell, who is nicknamed The Big Show and has played for Yorkshire as an overseas player this summer, displayed some fantastic reverse-sweeps with the bat as well as also launching Adil Rashid for two huge sixes.
However, he saved his show-stopping best for the end of England's run chase when he produced an incredible boundary catch to end Plunkett's entertaining late cameo.
He caught the ball with both feet off the ground but, aware he was going to step over the rope, flicked the ball up with his right hand and then caught it in his left as he re-entered the playing area in front of the Western Terrace.
His athletic best could not deny England though, and it is Morgan's side who have the momentum heading back to Old Trafford on Sunday.
Eoin Morgan inspired England to their best one-day international run chase against Australia as they levelled the five-match series to set up a decider at Old Trafford on Sunday.