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The skipper's 68 off just 48 balls, including two sixes and seven fours, was the heart of New Zealand's imposing 201 for four and although England had a bold stab at a target of 202 it proved just beyond their reach.
Defeat meant England had again lost by five runs after going down by that margin in a Champions Trophy final against India reduced to 20 overs per side by rain at Edgbaston on Sunday.
England needed six for a tie off the last ball that would have taken this match into a 'super-over' and that proved beyond Ravi Bopara, who finished on 30 not out.
The hosts fielded just four players from Sunday's side in stand-in captain Eoin Morgan, Bopara, Jos Buttler and James Tredwell.
McCullum shared a second-wicket stand of 114 with opener Hamish Rutherford, man-of-the-match for his dashing 62, after Boyd Rankin struck early on his England debut to leave New Zealand one for one.
After Morgan won the toss, it took fellow Irishman Rankin just four balls to take a wicket when James Franklin was out for a duck, wicket-keeper Buttler holding an inside edge.
Brendon McCullum saw a top-edge off Rankin fly 83 yards for a six over Buttler's head and Rutherford was even luckier when, on 25, a mistimed cut off Jade Dernbach's first ball was dropped by Bopara at point.
Rutherford used his reprieve to slog-sweep and drive off-spinner Tredwell for two sixes in as many balls.
But, going for another big hit, off medium-pacer Wright, he holed out having faced just 35 balls including four sixes and six fours.
Wright, who led the bowling with two for 31, struck again when he yorked Brendon McCullum before Ross Taylor's rapid 32 not out helped New Zealand beyond 200.
England openers Michael Lumb (29) and Alex Hales (39) posted a fifty stand inside four overs before Lumb played on to off-spinner Nathan McCullum.
However, the runs kept coming from a side still missing star batsman Kevin Pietersen, set to return to England duty in the second match of this series at The Oval on Thursday.
Brendon McCullum's move to bring a slip in as late as the 14th over was rewarded when Taylor took a brilliant leaping one-handed catch off a fiercely edged cut by Morgan as the dangerman fell for just seven off paceman Ian Butler.
And when Wright, who made 52 off just 34 balls to complete a fine all-round match, was caught behind by Tom Latham, off Mitchell McClenaghan, England were 139 for four.
England needed 26 off the final 12 balls under the floodlights in front of a capacity crowd of 23,500.
Buttler then produced his trademark 'ramp' shot for a six off Butler but, three balls later, missed with the same stroke and was run out after Latham, who didn't take his gloves off, hit the bowler's end stumps with an underarm throw.
England needed 16 off the final over, bowled by Corey Anderson.
Ben Stokes promptly hoisted Anderson for a huge legside six to make the target 10 off five balls only for the bowler to hold his nerve.
