With the series all square at 2-2 after an orgy of runs where totals under 350 had seemed below par, England reduced New Zealand to 261 for eight heading into the last over of the World Cup finalists' innings.
But the 50th over saw fast bowler Steven Finn concede 22 runs, including three sixes.
Wheeler, who only faced 28 balls in total, ended the innings by striking Finn for a four and two sixes off successive deliveries.
All-rounder Ben Stokes led England's attack with three for 52 in a maximum 10 overs on his Durham home ground.
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Leg-spinner Adil Rashid, whose eight overs in England's series-levelling seven-wicket win at Trent Bridge on Wednesday cost an expensive 75 runs, bounced back well with two for 45 in 10 overs.
New Zealand opener Martin Guptill (67) and the in-form Kane Williamson (50) both posted half centuries, while Ross Taylor, dropped on 19, made 47.
Their total did see World Cup finalists New Zealand exceed their previous highest aggregate for a five-match ODI series of 1,460 runs set against India in 2014/15.