New Zealand openers Tom Latham and Devon Conway departed cheaply before tea, and the crisis deepened when Kane Williamson and Tom Blundell were dismissed in successive balls in the final session with the hosts still in arrears.
Williamson offered hope with an innings of 61 but it all but disappeared when Chris Woakes, belying his reputation as a bowler who struggles outside England, trapped him in front and then had Blundell caught behind off a nick for a golden duck.
Glenn Phillips fended off the hat-trick ball but followed his teammates off the field in the last 20 minutes of play when he was dismissed lbw for 19 by Brydon Carse.
Daryl Mitchell, unbeaten on 31, and debutant all-rounder Nathan Smith, who had scored one not out, were at the crease at stumps with New Zealand just four runs ahead.
The Black Caps will have a long, hard look at their fielding as they reflect on their poor position after five dropped catches helped Brook to his match-turning tally.
England skipper Ben Stokes also benefited from a few as he provided the glue for a series of partnerships that pushed the visitors to 499 in response to New Zealand's 348.
Brook had resumed on 132 with England 5-319 and the 25-year-old played with great freedom and some style as he and Stokes put on 159 for the sixth wicket.
Defying the new ball, Brook punched his 13th four through covers to move past the 150 mark before blasting a huge six on to a grandstand roof.
By the time he got an edge on a Matt Henry delivery and was caught behind, Brook had hit 15 fours and three sixes in his 197-ball knock to take his batting average in three Tests in New Zealand to 100.
Henry (4-84) also finally dismissed Stokes for 80 but England's tail wagged hard with Gus Atkinson adding 48 runs off 36 balls and Carse an unbeaten 33 from 24.
Woakes (3-39) had failed to take a wicket in the first innings but wasted no time in removing Latham for one before Carse (3-22) and Atkinson combined to send back Conway for eight.
Williamson became the first New Zealander to accumulate 9,000 Test runs on his way to his 61, a rare highlight on a largely dismal day for the home fans.Â
