If New Zealand win 2-0, they will move to 109 points in second position and Sri Lanka will drop to seventh position on 90 points.
However, Pakistan's three dropped catches, two of them comparatively hard, would hurt them if New Zealand go on to draw the Test and walk away with a drawn three-match series.
Henry Nicholls was twice lucky to remain unbeaten on 20 off 54 deliveries at lunch on the fourth day in Abu Dhabi.
Ross Taylor's (22 off 14 balls) over aggressive approach ultimately led to his downfall when he was caught in the deep as New Zealand slumped to 4-60.
But the day started with Yasir's world record.
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Sarfraz Ahmed's side nevertheless regained control of the match by picking up two quick New Zealand wickets to leave them in trouble at 26-2 in 14 overs at close of play.
It was a remarkable comeback after they were bamboozled by Yasir Shah's legspin in the second test in Dubai. But BJ Watling resisted with 77 not out, delaying his record until the second innings.
Yasir took eight wickets in the opening Test against New Zealand but was unable to prevent the Kiwis from clinching a narrow victory.
That was followed by 24 wickets against Sri Lanka in 2015 and 10 in a match to beat England at Lord's a year later.
In fact, this century, the Black Caps had only claimed away test series victories over Zimbabwe, Bangladesh, and the West Indies, and considering that Pakistan had trounced Australia in their previous test series, victory should indeed be recognised as a superb accomplishment.
Shah already has 27 wickets in the series and is likely to break another record in this Test match, the most wickets in a three-Test series, a record which is now held by former Pakistan star Abdul Qadir who took 30 wickets against England in 1987-88.