At tea on day three in Christchurch, the hosts were 62-2 in their second innings after eventually ending England's first innings at 499.
Kane Williamson was on 26 and Rachin Ravindra not out 23 with the pair adding 39 runs as they worked to stitch together a big partnership.
Tom Latham was caught by Brook off Chris Woakes for one and Devon Conway fell for eight, collected by Gus Atkinson off the bowling of Brydon Carse.
In a rollicking day for England, they resumed at 319-5 and raced along at six an over before being all out after lunch.
Brook, who benefitted from five of the eight catches spilled by New Zealand, led England's salvage operation after they were left reeling at 71-4.
Captain Ben Stokes made 80 while Atkinson and Carse added spirited cameos with innings of 48 and 33.
The only good news for New Zealand was that while they dropped two more catches, taking the innings total to eight, they successfully held five.
Brook relished his lives. He smiled and looked to the heavens when Glenn Phillips spilled a chance on 147, having been dropped the previous day on 18, 41, 70 and 106.
Four overs after his last reprieve, he brought up his 150 in imperious fashion, charging down the wicket at Tim Southee with the second new ball and driving him to the cover boundary.
But when New Zealand found a way to make catches stick, Brook's innings, which included 15 fours and three sixes, came to an end.
A good length ball from Matt Henry found a faint outside edge and wicketkeeper Tom Blundell grasped the opportunity.
The applause as Brook left the ground was as much for New Zealand holding a catch as it was for a magnificent innings.
Woakes (one) was out on the second ball he faced with Latham, who had earlier dropped three offerings, scooping up a low-level chance at second slip.
Nathan Smith ended Atkinson's breezy innings with Phillips taking a smart catch at long leg.
Henry was the most successful New Zealand bowler with 4-84. Smith took 3-141.
