Sri Lanka were 110 for five at lunch, chasing a victory target of 390, after Boult removed nightwatchman Dhammika Prasad (six) and Kumar Sangakkara (five) in quick succession early in the session.
Bracewell then struck twice before lunch, removing opener Kaushal Silva, who had just scored his seventh test half century but was then caught by Mark Craig in the slips, before Angelo Mathews was caught for eight by Kane Williamson in the gully.
Lahiru Thirimanne went to the break on 11, with the visitors facing a tough task to save the match let alone secure a series-levelling victory.
Much of their hopes had rested on the shoulders of Sangakkara, who scored a brilliant 203 in the first innings, but he fell in controversial circumstances when he was initially given not out before New Zealand asked for a review.
Technology indicated that Sangakkara got a faint edge to a Boult delivery and the original decision was overturned, leaving the veteran batsman to wander off shaking his head.
Prasad, who had come to the wicket late on Tuesday after Craig dismissed Dimuth Karunaratne for 17, did not hang around long on Wednesday when Jimmy Neesham took a sharp catch in the slips off Boult to bring Sangakkara to the wicket.
The hosts, who won the first game in the two-test series by eight wickets, have completed a remarkable turnaround in the match that they had looked likely to lose inside three days.
They had slumped to 159 for five, a lead of 24 runs in their second innings on Monday, before Williamson and BJ Watling combined for a world record sixth-wicket partnership of 365.
Their partnership surpassed the 352 Watling and Brendon McCullum scored against India last year, also at the Basin Reserve.
