Wicketkeeper Watling was unbeaten on 42 at stumps, hitting only one boundary in 180 balls, with debutant William Somerville keeping him company on 12.
Leg-spinner Yasir, who took 14 wickets as Pakistan won the second test in Dubai last week, claimed three wickets in the morning after Williamson won the toss and elected to bat.
After fast bowler Shaheen Shah Afridi dismissed Tom Latham lbw for four, Shah removed Jeet Raval (45) with a flat delivery that caught the opener plumb in front of the wicket.
The 32-year-old Shah, who began the match needing five wickets to replace Clarrie Grimmett as the fastest man to claim 200 test victims, bowled Ross Taylor with the next ball to inch closer to the landmark in his 33rd match.
Australian leg-spinner Grimmett reached 200 wickets in his 36th test in 1936.
Although Shah missed out on a hat-trick, he struck again to bowl Henry Nicholls around his legs and leave the visitors in trouble at 72-4.
Williamson and Watling then dropped anchor, sharing a 104-run fifth-wicket stand from 249 balls.
The Kiwi skipper, who hit seven boundaries and looked set to reach his 19th test century, perished when he was caught at short mid-wicket off paceman Hasan Ali.
Off-spinner Bilal Asif dismissed Colin de Grandhomme for 20 and Tim Southee for two late in the day to give Pakistan the advantage.
New Zealand won the opening test in Abu Dhabi by four runs before Pakistan levelled the three-match series with victory in the second test in Dubai by an innings and 16 runs.
(Reporting by Shrivathsa Sridhar in Bengaluru, editing by Ed Osmond)
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