The pair shared a smooth second-wicket partnership of 185 in excellent batting conditions at Old Trafford to frustrate the Pakistanis who won the first Test at Lord's.
Cook was bowled by Mohammad Amir for 105 on the stroke of tea but Root continued to flay the tiring attack to all corners of the ground and finished unbeaten on 141 with Chris Woakes two not out.
Pakistan started well after losing the toss when fast bowler Amir bowled Alex Hales for 10 with a rapid full-pitched delivery.
It allowed him to equal Australian great Don Bradman's tally of Test centuries and tie Essex mentor Graham Gooch's record total of 11 hundreds as England Test captain.
Amir caused a surprise when he produced a ball that kept low and crashed into Cook's stumps off the bottom of the bat.
James Vince played some well-timed strokes on his way to 18 but, as so often before in his fledgling Test career, he was undone by a full ball outside off stump and edged Rahat Ali to wicketkeeper Sarfraz Ahmed.
Gary Ballance then made a careful 23 before he chopped a lifting ball from Rahat on to his stumps.