Aslam and Azhar Ali (58) produced a 131-run stand for the first wicket, and with Babar Azam not out on 16 the tourists were set for a final showdown at Hamilton's Seddon Park.
It is not a foreign situation for them.
When they required 292 in 62 overs at lunch it evoked memories of an epic run chase in Sharjah two years ago when they smacked 302 in the fourth innings in 57.3 overs to beat Sri Lanka.
Despite the Seddon Park wicket still offering some support to the bowlers on a flattening pitch, Azhar and Aslam were able to set the stage for another grandstand finish.
Pakistan require 211 runs off 204 balls if they are to snatch victory and level the series.
A draw or a loss will consign them to fourth place below England and Australia.
At the top of the innings, Azhar and Aslam were resolute with their best performances of the series although Aslam had two heart-stopping moments.
On 31 he was given not out on an lbw appeal when Kane Williamson debated too long on whether to go for a referral and his chance to request a review timed out. Replays showed Aslam was plumb in front.
He could also have been run out on 45 had Mitchell Santer's throw from midwicket been fast and direct to the wicketkeeper.
Aslam brought up his sixth 50 in nine Tests carving a Neil Wagner bouncer to the boundary, giving him some satisfaction in a match where he dropped three critical catch offerings in the field.
Azhar, standing in as captain for the unavailable Misbah-ul-Haq, faced 147 deliveries before he reached his 23rd half century with a single off Matt Henry.
But his patient innings ended soon after when he went to drive Santner out of the ground only to drag the ball back on to his stumps.
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