England, with a commanding 2-0 lead in the series, find themselves 204 runs behind as they bowled out the Black Caps for 347 on a scorching day at Seddon Park.
Henry removed the England openers in the fifth over of the innings, taking a brilliant low catch to remove Zak Crawley for 21 before trapping Ben Duckett plumb in front for 11 four balls later.
O'Rourke stepped up after lunch with two wickets in two balls, having Jacob Bethell caught at point for 12 before danger man Harry Brook chopped on for a golden duck.
Pope saw off the hat-trick ball but Joe Root had no such luck when an O'Rourke delivery reared up into his face and the batsman cut it to Will Young at gully to depart for 32 with England having lost three wickets for five runs.
Pope and Ben Stokes put on 52 for the sixth wicket to rescue England from 82-5 but spinner Santner came on for the first time and had the England wicketkeeper caught in the slips for 24 with his fifth delivery.
The left-armer also sent back the England captain lbw for 27 in his next over before he and Henry tag-teamed to remove Brydon Carse, Gus Atkinson and Matthew Potts cheaply as the England tail folded.
Santner and O'Rourke had earlier added 32 runs to New Zealand's overnight tally before Matthew Potts ended their last-wicket stand on 44 by bowling the spinner for 76.
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