Root began the day on 32, needing a further 39 runs to eclipse Cook's tally of 12,472, and the former captain eased to his target with an on-drive for four off Aamer Jamal before soaking up loud applause from the travelling English fans.
He then fought off cramps after lunch to bring up his 35th century with a reverse-sweep and looked poised to plunder more runs after cruising to 176 not out at the close with England trailing by only 64 runs.
England's Harry Brook also got in on the high-scoring act with his own ton. (AP PHOTO)
Fellow Yorkshireman Harry Brook was still batting on 141, having shared an unbeaten 243-run stand for the fourth wicket.
Brook's fine knock followed Ben Duckett's quickfire 84 earlier as England put themselves in a fine position to attack on the fourth day and put the hosts under pressure.
Pakistan's bowlers were unable to extract any help from the flat pitch at the Multan Cricket Stadium and were wrestled into submission by England, who had toiled on the first two days of the match.
Root got to his fifty earlier with an inside edge and the 33-year-old survived a loud lbw appeal to go past Cook's record and into the top-five in the all-time list behind Rahul Dravid, Jacques Kallis, Ricky Ponting and Sachin Tendulkar.
Resuming on 1-96 after Pakistan amassed 556 in their first innings, England lost Zak Crawley for 78 inside the first hour of the day when the opener flicked one from Shaheen Afridi to Aamer Jamal at midwicket.
That abruptly ended the 109-run stand for the second wicket but England were in no mood to slow down.
Duckett, who dislocated his left thumb while taking a catch and was unable to open on Tuesday, showed no signs of the issue as he took to the crease and smashed spinner Abrar Ahmed out of the attack temporarily with three boundaries in the 30th over.
The left-hander needed only 45 balls to bring up his half-century but Aamer trapped him lbw in the afternoon session to snap his 136-run stand with Root and leave England on 3-249.
There would be no respite for Pakistan as Brook ensured the run rate did not drop below five an over at tea, with his breezy effort helping England post their third consecutive century stand of the innings.
To add to the hosts' misery, Brook bizarrely survived a scare in the evening session when the ball hit the stumps after bouncing off his helmet grill and the bails did not fall off.
Brook quickly shrugged off that incident to complete his sixth Test ton.
