England last week went 2-0 up in the three-match series playing bold and aggressive cricket under their new leadership pair of captain Ben Stokes and coach Brendon McCullum.
But England lost six wickets within 12 overs, mostly due to good bowling by the Test world champions, rather than cavalier batting.
Bairstow and Overton, however, counter-attacked in fine style, although the Yorkshireman was missed on 27 when recalled paceman Neil Wagner dropped a return catch off a checked drive.
That was a rare error by Bairstow, whose blistering 136 had been the cornerstone of England’s successful run-chase in a five-wicket and series-clinching win at Trent Bridge.
Overton, selected ahead of twin brother Craig in place of the injured James Anderson, demonstrated a wide array of attacking shots.
His 68-ball fifty featured seven fours and a six, with the 28-year-old following up by pulling and driving Wagner for six and four off successive deliveries.
Bairstow, having reached fifth in 51 balls, completed his 10th hundred in 86 career Tests by off-driving Boult for a 15th four in 95 balls faced.
The 32-year-old son of the late England wicketkeeper David Bairstow sprinted in celebration towards the stand where his mother, Janet, was watching before stopping short of the boundary and acknowledging the crowd’s applause.
Earlier, left-hander Alex Lees was bowled off the fifth ball of England’s reply by a superb Boult leg-cutter before the left-arm quick cleaned up Ollie Pope and Zak Crawley.
Southee then had England star batsman Joe Root caught behind for five to leave England reeling at 21-4.
Although Stokes counter-attacked by driving Southee for six, he could only chip Wagner to mid-off attempting a similar shot.
Three balls later, Wagner in for the injured Kyle Jamieson, had a second wicket in only his first over of the series when Ben Foakes was plumb lbw for nought to leave England 55-6.
New Zealand had resumed on 225-5, with Mitchell turning his overnight 78 not out into an innings of 109.
In the process he broke a 73-year-old New Zealand record for the most runs during a series against England, with Mitchell now having compiled 482 this campaign — 20 more than Martin Donnelly managed in 1949.
Mitchell put on 120 with Tom Blundell, who made 55 before he was lbw to Matthew Potts.
But Mitchell completed his latest hundred in fairytale fashion by lofting Jack Leach over long-off for six.
Trying to repeat the stroke, however, his 228-ball innings ended when a miscued drive off the spinner was well caught by Stokes, running back from mid-off.
Leach, whose Test career has been blighted by illness, injury and inconsistent selection, finished with an impressive of 5-100 in 38.3 overs.
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