A battling 90 from Kane Williamson and half centuries from Grant Elliott and Martin Guptill had helped the tourists reach 349-7, but Alex Hales got England off to a flying start with a quickfire 67 before Morgan and Root took over.
A remarkably high-scoring series will now be decided in Durham at Chester-le-Street on Saturday.
"It's certainly up there with one of the best games of cricket I've ever played in. The boys showed that aggressive intent right from the get-go," Morgan said on the BBC's Test Match Special.
Brendon McCullum won the toss and chose to bat with New Zealand making steady progress before Williamson came to the crease at 88 for one.
He pushed New Zealand's score past 200 with another century partnership with Ross Taylor (42), while also passing 3000 ODI runs.
He fell 10 runs short of a second consecutive century after he mistimed a drive and was caught by Adil Rashid.
Rashid was in the thick of the action at the end of the tourists' innings, conceding 28 runs in his penultimate over as New Zealand sought to build an imposing total with some late hitting.
A new-look England side, however, enthused with a new aggressive outlook, swiftly set about chipping away at the total.
Hales got England off to a flying start, smashing 67 off 38 balls, before he dragged on trying to heave Matt Henry down the ground while fellow opener Jason Roy (38) followed soon after to leave the hosts on 111 for two.
Morgan and Root swiftly restored the momentum, however, with the captain racing to his eighth ODI hundred off 73 balls before he was eventually caught by Henry at fine leg with England closing in on victory on 309 for three.
Ben Stokes arrived at the crease to help Root reach three figures, before England swiftly wrapped up proceedings.
It was a dispiriting result for New Zealand captain McCullum, who was confident his side had done enough with their first innings total.
"It was an incredible run chase," he said.
"We knew they were going to have to play well to beat us and play well they did. They blew us off the park again."
(Reporting by Toby Davis in London; editing by Pritha Sarkar)
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