Bell was unbeaten on 40 and was ably supported by a resolute James Taylor (26) as England, 135-3 at tea, trail Pakistan's first innings total of 234 by 99 runs with seven wickets intact at Sharjah stadium.
England need a big lead in their hunt for a series-levelling win after losing the second Test in Dubai by 178 runs and drawing the first in Abu Dhabi.
Bell, who made 4 and 46 in Dubai, has so far hit two boundaries and a six during his 151-ball knock, holding the innings together.
England had added three runs to their lunch score of 87-1 when Cook fell to a soft dismissal as he pushed a Yasir Shah delivery straight into the hands of short leg fielder Azhar Ali.
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Cook hit four boundaries during his 119-ball knock.
England had hoped Root would supplement Bell but the fast-rising batsman fell to a loose shot off paceman Rahat Ali and was smartly snapped low to his right by wicketkeeper Sarfraz Ahmed.
Resuming at 4-0, England lost Ali in the seventh over of the day when he miscued an aggressive shot off spinner Shoaib Malik and was caught in the slip by Younis Khan for 14.
Cook, who made an epic 263 in the drawn first Test in Abu Dhabi, looked in no trouble as he negotiated Pakistan's pace-cum-spin attack with confidence, looked set for another half-century before he fell.