England captain Cook was unbeaten on 46 and Root was 45 not out as they lifted England from a precarious 14-2 to 105-2 at tea in an unbroken third wicket stand of 91.
That left England 273 behind Pakistan's first innings total of 378 with eight wickets intact.
The first Test in the three-match series ended in a draw in Abu Dhabi last week.
Pakistan seamers Wahab Riaz and Imran Khan made early inroads to dismiss opener Moeen Ali (one) and Ian Bell (four) but spinners Yasir Shah and Zulfiqar Babar failed to make an early impact.
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Cook survived a confident leg-before appeal off spinner Zulfiqar Babar on 27 as his attempted sweep hit his pads and the ball deflected on to the stumps and despite having an impact the bails were not removed.
Pakistan also unsuccessfully reviewed a leg-before shout when Cook edged Riaz's delivery on to his pads when on 41.
Earlier, Pakistan, resuming at 282-4, lost their last six wickets for 96 runs in the two-and-a-half hour session, extended due to Friday prayers.
Asad Shafiq scored 83 before he was caught by Root off Wood, 17 short of his hundred.
He hit nine fours off 178 balls and valiantly attempted to reach his century but he was running out of partners.
Pakistan were jolted when skipper Misbah-ul-Haq was dismissed off the fifth ball of the day, trapped leg-before by paceman Stuart Broad.
Misbah, who had not added to his overnight score of 102, reviewed Australian umpire Paul Reiffel's decision but the replays showed the ball was hitting the leg-stump.
Misbah hit eight fours and five sixes during his 197-ball stay at the crease.
Shah scored a quick 16, including three fours.