The pair, who put on 185 for the second wicket, helped the hosts to 314 for four at stumps as they seek a winning response to their opening defeat at Lord’s.
Cook (105) cashed in first – after winning the toss in sunny conditions, on a pitch of even pace and bounce – and Root (141no) followed him to three-figures.
Both chanceless hundreds, crucial contributions at a pivotal point in the four-match series, had added personal value.
Cook’s 29th, in his 50th Test as captain, equalled the number made by the great Don Bradman and was also his first in 20 innings dating back to the tour-de-force 263 against these same opponents in Abu Dhabi nine months ago.
For Root, century number 10 was his first of a summer in which he has – by his own admission – squandered several opportunities.
It was notable for a painstaking eradication of the errors which have interrupted his progress of late, especially a determination to let anything wide whistle by and wait for Pakistan to put the ball in his risk-free scoring zones.
He and Cook achieved command of the attack which had proved too much for England at HQ, the captain profiting with a 157-ball hundred completed shortly before his dismissal to Mohammad Amir on the stroke of tea.
Cook started his innings with a moment of fortune, edging the final ball of Amir’s first over at catchable height through a vacant fourth slip to double his score.
Misbah-ul-Haq promptly added the extra slip, too late.
Amir did work over Cook’s opening partner Alex Hales, though.
A series of outswingers included one squirted for four through gully’s fingertips before, in the same over, Amir brought one back brilliantly to beat Hales’ defence and knock out off-stump.
Root began imperiously with a straight-drive for four off Rahat Ali third ball, quickly closed on Cook and narrowly beat him to 50 in early afternoon.
Cook has rarely been more fluent in his record-breaking career, though, and soon established a lead on his partner again – leaning heavily on his favoured cut shot but also picking up boundaries with check-drives poked down the ground off both pace and spin.
Amir had to run the gauntlet of local humour by the time he got into his third spell as the Manchester crowd demonstrated their memory of his spot-fixing past with a succession of ironic ‘no-ball’ calls from the temporary stand square with the wicket.
He did not overstep once, in fact – and it was from one of his legitimate deliveries that Cook completed his hundred with a two clipped wide of mid-on.
The England captain celebrated his milestone with mildy un-Cook-like gestures of aggression – including a pumped fist as he approached the striker’s stumps for the telling second run.
Amir got his revenge when Cook went back to a length ball and got an under-edge onto his stumps, and in early evening James Vince was unable to take advantage of a let-off at second slip – going after another wide drive and edging Rahat behind.
It was the latest instalment in a summer of groundhog experiences as Vince struggles to establish himself in the middle order.
But Root was rewarded for his skill and patience when his 14th four, an on-drive off Yasir Shah, took him to his hundred from 167 balls.
Yasir, on top of the world rankings after his 10-wicket haul at Lord’s, discovered it will not always be so easy in England – and long before stumps had conceded more than 100 runs, without success.
Amir was the pick of the Pakistani attack, but Root remained in charge – and as if to press the point, duly pulled the left-armer’s first delivery with the second new ball for four.
Gary Ballance had to work hard for his runs, and eventually chopped on to Rahat, but his and Root’s stand of 73 still put the seal on England’s day.
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