England captain Eoin Morgan made 71 on a ground where he averages 70.5 in eight one-day international innings.
And all-rounder Ben Stokes made a dashing 68 off 47 balls.
But England lost their last five wickets for 14 runs, with 28 balls of the innings still left when Steven Finn was last man out.
Southee finished with three for 44 in 8.2 overs and Wheeler, replacing Trent Boult after his fellow left-arm paceman was ruled out of the rest of the tour with a back injury, an impressive three for 63 in 10 overs.
Together with Joe Root (54), he put on 105 for the third wicket after England had lost two wickets for no runs in three balls to be 34 for two.
Morgan opted to bat first after winning the toss on Sunday.
It was an unsurprising decision given England had piled up a record 408 for nine during a 210-run win after being sent into bat in the series opener at Edgbaston, while New Zealand levelled the five-match contest at 1-1, albeit under the Duckworth/Lewis method, after making 398 for five at The Oval.
And 34 for one became 34 for two when Jason Roy was bowled by a Southee nip-backer.
Root and Morgan repaired the damage.
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