James Neesham (180), Neil Broom (11) and Correy Anderson (10) tried hard to dig New Zealand out of the trouble but their contributions were not good enough.
Chasing 311 for victory, New Zealand lost the plot after in-form skipper Kane Williamson (87) departed at a crucial juncture and could only manage 223 all out in 44.3 overs.
Fortunately, Williamson was sufficiently recovered to put up his own fifty, as New Zealand edged toward a collective century with a little more than 28 overs left.
"With the inconsistency of the pitch, I thought if we genuinely did take one (wicket), we would take two", Morgan said. But rain interrupt the innings.
It was the golden arm of Stokes (1-46), though, which brought England's second breakthrough, with Martin Guptill (27) driving him to Root at slip a ball after drilling him down the ground for four. New Zealand were now 63-2 and relying on Ross Taylor to pick up the slack.
But just when it seemed the duo were in course to take New Zealand ahead, Williamson edged one to Root behind the stumps off the bowling of Mark Wood (1/32). The half-century from the classy right-handed batsman came in 52 deliveries.
Williamson was disappointed with his side's performance, and said: "They bat very deep with a lot of power, so restricting them wasn't a bad effort".
"But the fact we're still trying to win games with the bat, regardless of the situation that we're in, I think is testament to the team".
It was one of those unusual days.
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Two years ago, England's woeful first-round exit from the last International Cricket Council 50-over tournament included a humiliating eight-wicket thrashing by New Zealand in Wellington. Stokes proved a key threat with a knock that included four fours and two sixes before he looped an uppercut nearly straight to Milne at third man off Boult.
Magnificent maximums! WATCH a compilation of England's sixes in their win over New Zealand_!
To add insult to injury the Blackcaps were also fined for a slow over-rate, Williamson docked 40 per cent of his match fee and the rest of the team 20 per cent each.
Victory for England on Tuesday would see them into the semi-finals of a 50-over tournament featuring the world's top eight one-day worldwide teams. India has two points in its account after thrashing arch-rivals Pakistan on 4 June.
Currently England is on the top of Group A points table with 2 points and +0.407NRR and New Zealand is on 3rd position with 1 point. The Kiwis will now have to beat Bangladesh in their last group A encounter and hope that England beat Australia.
Morgan promised no let-up against the Australians.
He said: "I certainly think [we have alighted on our best team]".
The tournament hosts will now return to the Welsh capital next Wednesday as Group A winners - even if they lose to Australia at Edgbaston on Saturday - to face the runners-up in Group B.