With the pair at the crease, Pakistan were set to threaten the Eden Park ODI innings record of 340, but with their departure the innings collapsed and they were unable to bat out their 50 overs.
Pakistan, who need to win to tie the series, were all out for 290 with 15 balls remaining with Babar contributing a career-best 83 and Hafeez 76.
The tourists were 20-2 in the sixth over when the pair embarked on a 134-run partnership for the third wicket, in which they lifted the run rate to 6.81 by the time Hafeez was dismissed with 27 overs remaining.
Martin Guptill, who was in doubt on the eve of the match because of a twisted ankle, was instrumental in the first two wickets that fell.
Fielding at short midwicket, he took a smart catch low down to give Boult the wicket of Ahmed Shehzad for 12.
Two balls later, fielding at slip to Henry, he caught Azhar Ali for three and Pakistan were quickly two down after winning the toss and electing to bat.
Hafeez eventually fell to Mitchell Santner, caught by Adam Milne at deep backward square leg, for 76 which included five fours and five sixes.
It was a satisfying wicket for Santner in his third over after his first two cost 23 runs.
Malik maintained the onslaught with Babar, adding a rapid 32 off 27 until he was caught behind to give Boult his second wicket.
Babar, whose 83 off 77 contained 10 fours and a six, was removed two overs later when he slashed a Henry delivery to the safe hands of Guptill at point.
For New Zealand, Milne took three for 49 while Boult and Henry finished with two apiece.