Cook continued to bat without much problem as he reached 116 to take his team to 197 for one at tea with Ian Bell on 31 as the pair thwart Pakistan's pace-cum-spin attack on a Sheikh Zayed Stadium pitch.
England still trail Pakistan's mammoth first innings total of 523 for eight by 326 runs.
Cook drove left-arm fast bowler Wahab Riaz for an exquisite cover drive for his tenth boundary to complete his 28th century in Tests, raising his bat in celebration.
He has so far hit 11 boundaries during his 222-ball stay at the wicket.
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Pakistan thought they had England skipper leg-before when he failed to connect a sweep off left-arm spinner Zulfiqar Babar but Australian umpire Paul Reiffel's decision of not out was upheld on review.
Earlier, Cook and Moeen Ali resumed the day at 56-0, bringing up the century stand in the first hour of play as Pakistan attacked through pace at one end and spin from the other.
But at 116, left-hander Ali edged an outgoing delivery from paceman Imran Khan to wicket-keeper Sarfraz Ahmed who took a low catch to his left.
Bell survived some anxious moments at the crease with left-armer Zulfiqar Babar started to turn the ball on a pitch which did not assist England spinners on the first two days.
Babar was unlucky not to have Bell four overs before the interval when close-in fielder Shan Masood failed to hold on to an edge at silly point.