At 24, Babar Azam reached his landmark in 68 innings; while this was 11 innings more than South Africa's Hashim Amla (57), it was faster than the legendary Viv Richards, who achieved it in 69.
Pakistan found themselves at 110-3 after Hafeez was dismissed for 32, caught by Lockie Ferguson off the bowling of Kane Williamson. Meanwhile, New Zealand holds second place in the rankings with 11 points, just behind Australia, the only team to have already qualified.
The semi-final berths are reserved for the top four teams in the standings.
In the opening half of the game, New Zealand finished with 237-6, highlighted by a standout, career-best 97 not out from Jimmy Neesham and a contribution of 64 by Colin de Grandhomme.
Shaheen Shah Afridi was the standout bowler for Pakistan, taking 3 wickets for 28 runs and leaving New Zealand reeling at 83-5.
New Zealand's momentum collapsed when the dangerous Kane Williamson was tricked by leg-spinner Shadab Khan, edging the ball to wicketkeeper Sarfaraz Ahmed for 41.
Despite being second in the 10-team league with 11 points, New Zealand opted to field the same lineup for the fifth consecutive game.