Pakistan, who lead the three-match series 1-0, were grateful to Sarfraz as they progressed from an overnight score of 281-6 to reach 393 all out in Dubai and concede a slender first-innings lead of 10.
The wicketkeeper-batsman compiled 112, sharing 81 for the 10th wicket with Rahat Ali (16).
Zulfiqar Babar (3-60) and Yasir Shah (3-65) then shared six wickets as New Zealand stumbled to 167-6 despite an admirable innings of 77 not out from Ross Taylor.
As a result, the match is delicately poised heading into the final day, with the Black Caps 177 ahead and hopeful of further defiance from Taylor as they aim to at least keep the series alive.
Sarfraz resumed on 59 and held firm despite seeing Yasir (two), Ehsan Adil (nought) and Zulfiqar (five) fall cheaply in the morning session.
Rahat ultimately proved a willing ally, batting for an hour and 45 minutes and enabling Sarfraz to bring up his third Test hundred.
The centurion fell to an unlikely wicket-taker, New Zealand captain Brendon McCullum accepting a return catch for his maiden Test scalp.
McCullum batted positively at the start of his side's second innings, striking six fours on his way to 45.
However, Tom Latham was caught at leg gully off Yasir for nine - a review failing to save the in-form opener even though the ball appeared to miss his glove - and three batsmen fell in the space of 17 balls after tea, including McCullum, to leave New Zealand in a perilous position at 79-4.
Jimmy Neesham and BJ Watling could only muster 11 runs apiece, the latter falling in the day's penultimate over, but Taylor prospered on the cut in a counter-attacking 93-ball innings.
The 30-year-old will surely be key to New Zealand hopes on day five, which he will begin alongside Mark Craig (nought).