England went 2-0 in the three-game series last week by playing daring and aggressive cricket under their new leadership pair of captain Ben Stokes and coach Brendon McCullum.
But England lost six wickets within 12 overs, mainly due to good bowling by the Test world champions, rather than cavalier batting.
However, Bairstow and Overton counterattacked handsomely, although the Yorkshireman was missed at 27 when recalled pacer Neil Wagner dropped a flyout from a controlled drive.
That was a rare mistake by Bairstow, whose blistering 136 had been the cornerstone of England’s successful run-chase in a five wicket run at Trent Bridge.
Selected for twin brother Craig in place of the injured James Anderson, Overton demonstrated a wide range of attacking shots.
His 68-ball fifty had seven fours and a six, with the 28-year-old following him by pulling and driving Wagner six and four of consecutive deliveries.
Bairstow, who had finished fifth in 51 balls, completed his 10th hundred in 86 career tests by knocking out Boult for a 15th four in 95 balls.
The 32-year-old son of the late England wicket-keeper David Bairstow sprinted to the stands in celebration where his mother, Janet, watched before stopping at the border and receiving the applause from the crowd.
Previously, lefthander Alex Lees was thrown off the fifth ball of England’s answer by a superb Boult leg-cutter before the left arm cleared Ollie Pope and Zak Crawley quickly.
Southee then left England star batsman Joe Root for five catches to stagger England at 21-4.
Although Stokes counterattacked by driving Southee for six, he could only break Wagner off halfway when attempting a similar shot.
Three balls later, Wagner in for the injured Kyle Jamieson, had a second wicket in just his first left of the series when Ben Foakes was in vain to leave England 55-6.
New Zealand had resumed at 225-5, with Mitchell not turning out his 78 on the night in an innings of 109.
In the process, he broke a 73-year-old New Zealand record for most runs in a series against England, while Mitchell has now put together 482 this campaign – 20 more than Martin Donnelly managed in 1949.
Mitchell set up 120 with Tom Blundell, who made 55 before being lbw to Matthew Potts.
But Mitchell magically completed his last hundred by elevating Jack Leach to six.
However, in an attempt to repeat the trick, his 228 ball innings ended when a misplaced drive from the spinner was well caught by Stokes, who ran back from the middle.
Leach, whose testing career was blighted by illness, injuries and inconsistent selection, finished an impressive 5-100 in 38.3 overs.
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