The duo shared 182 runs as the tourists reached an imposing 241-3 - England's highest partnership and innings total in the format.
His 103 included six sixes and nine fours while Morgan cracked seven sixes and seven fours.
In a devastating display of hitting from the former Middlesex team-mates, Morgan set a new benchmark for the fastest T20 half-century by an England batsman. Other opener Jonny Bairstow was dismissed for just 8 runs before that. If I play that's good and if not that's just the way this tour works.
The two sides are now closely matched once again, with the five-game T20 series level at 2-2 after England's 76-run win in Napier on Friday.
Malan is barely the second England participant to attain a T20I century after Alex Hales, who wanted 60 balls to achieve his hundred.
Mitchell was sacrificed to the batting gods in the penultimate over and watched on as another 25 runs were tacked on to the scoreboard, before Morgan finally holed out with two balls remaining - Mitchell this time clinging onto the catch that he couldn't 80 runs earlier, as the skipper departed for 91 from 41 balls.
In fairness, the New Zealanders could have bowled excellently and it may not have made a difference, such was the power and precision that Malan and Morgan were striking with.
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Mitchell Santner took both the wickets.
A sound middle-purchase partnership by Colin de Grandhomme and Ross Taylor laid the foundation for New Zealand immediately after Martin Guptill had given them a traveling commence with 33 off 17 deliveries.
Tim Southee, though, tried to bring some respite and smashed 39 runs in just 15 balls.
Of the New Zealand bowlers, Blair Tickner completed with none for 50 off his 4 overs, whereas Sodhi had none for 49 off three overs and Southee bowled his full complement to complete with one for 47. Chris Jordan got two, Sam Curran one, Tom Curran one and Pat Brown also got one wicket each.
This is also the second-highest total by any team against New Zealand, behind Australia's 245 for 5 in Auckland past year.
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