Pakistan were all out for 216 at stumps on the third day, trailing New Zealand by 55 as rain reduced the day’s play.
The 22-year-old, in only his third Test, was not out on 90, surpassing his previous Test best of 69 on debut against the West Indies last month. Mohammad Amir made five and Imran Khan made six before the visitors were dismissed.
"I'm thankful the lower order gave some resistance but it was unfortunate I could not complete my 100," said Azam after stumps.
In reply, New Zealand faced only one ball for no run in their second innings when play was washed out.
Pakistan in trouble in reply of New Zealand's 271
After starting the day on the ropes at 76 for 5, Pakistan added 140 during the day, which began 30 minutes early to make up time lost to rain on the first day.
Azam resumed on 34 and took the senior role of keeping the innings together while Sarfraz Ahmed (41) and Sohail Khan (37) attacked the bowling.
Sarfraz had luck on his side when he pulled Tim Southee to midwicket where the ball went through the hands of a leaping Henry Nicholl. Four balls later Tom Latham missed with a shy at the stumps as Sarfraz slipped midwicket after being turned back during an aborted single attempt.
New Zealand end first day’s play on 77-2
Bowling with a blustery wind behind him, Southee could not reproduce his commanding form of Saturday when he removed the Pakistan top order. However, he made an impressive comeback and claimed three more wickets during the day, finishing off with the figures of six for 80.
"There was still a bit of swing at that stage and I went back to what we've done well for a number of years," said Southee. "You've got plans you put in place for certain batsmen and it's nice when they do come off."
He added: "There's a little bit there for the new ball, but once the ball gets older it gets a little bit easier to bat but there will still be a little bit of assistance throughout."
Pakistani bowlers failed to exploit conditions, says former chief selector
Southee also praised Azam for his innings. "For a guy to come here in his first Test tour of New Zealand and play the way he's played is a good sign he can play in conditions that are foreign to them," he said.
The introduction of Neil Wagner produced the breakthrough for New Zealand. Sarfraz, who had slashed and profited against Southee, tried the same shot to a short Wagner delivery and it flew to Jeet Raval at second slip. Wagner took three for 59.
Wahab Riaz faced only five balls before he was lbw to Colin de Grandhomme.
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