The pair came together in Christchurch with England in dire straits at 71-4 after the New Zealand seamers had ripped out the top order.
They have so far added 103 for the fifth wicket, riding their luck with an aggressive approach. At the break Pope was on 59 and Brook 54.
Brook was dropped twice, by Glenn Phillips on 18 and Tom Latham on 41, with both batsmen having an element of luck on their side as several attempted drives became edges that flew past the slips cordon.
It was not a good day so far for New Zealand captain Latham, who also dropped Ben Duckett on 23.
Duckett added a further 23 runs before he was the only wicket to fall in the second session when he hooked a rising Will O'Rourke delivery to Devon Conway on the boundary.
England lost three wickets before lunch, including two in the final over when debutant Nathan Smith dismissed Jacob Bethell and Joe Root.
The 21-year-old Bethell was thrust into action in the fourth over when Zac Crawley was trapped lbw by Matt Henry without scoring.
Bethell faced 12 deliveries before getting off the mark with a single from Henry, who had Duckett dropped at second slip by Latham with his next ball.
When Smith was introduced to the attack, Bethell showed his promise with two boundaries in his first over.
But with the first ball of his second over, he had Bethell caught behind by Tom Blundell for 10 and then bowled Root for a duck on the former England skipper's 150th Test.
New Zealand resumed on 319-8 and added a further 29 runs, the bulk of them by Phillips.
He ended unbeaten on 58, his fifth half-century and the second-highest New Zealand score behind Kane Williamson's 93.
Brydon Carse finished with 4-64, the best figures for England, after taking both wickets to fall on the second morning.
With his first ball of the day he had Tim Southee caught on the boundary for 15, then ended the hosts' innings by bowling O'Rourke for a duck.
