Brook made 111 to continue his excellent start to test cricket, with the batsman proving to be the glue in England's innings, putting on partnerships with Ollie Pope and captain Ben Stokes before a stand of 117 with Foakes.
Spinners Nauman Ali and Abrar Ahmed played a big role in stemming England's flow on either side of the sixth-wicket stand, picking up four wickets apiece as the visitors were eventually dismissed for 354, in response to Pakistan's first-innings total of 304.
England lost two wickets in quick succession shortly after resuming on 7-1, with Nauman first trapping Ben Duckett lbw and then getting Joe Root to depart for a duck after being caught in the slips on the next delivery.
Nauman was denied a hat-trick by Brook, who kept England's chase going, despite Pope being bowled by Abrar and Stokes losing his wicket in a run-out following a mix-up between the batters.
Brook and Foakes then revived England's innings, building a defiant partnership which ended when Mohammad Wasim was able to find a chink in Brook's armour to dismiss him lbw.
"It's been a good day for us today. We tried to put it back on them and get a bit of a lead," Brook told Sky Sports.
Foakes valiantly fought on after Brook's departure, scoring 64, but skied a delivery from Nauman to mid-on, where it was caught by Abdullah Shafique.
The last bit of English resistance came courtesy of Jack Leach and Ollie Robinson, who combined in a 30-run stand before Robinson was bowled by Abrar to end England's innings.
England were unable to take any wickets in the few overs that Pakistan batted on day two, with Shafique ending on 14 while Shan Masood was on 3.
"It would have been nice to have got a couple of wickets tonight but I'm sure we will go hard at them tomorrow," Brook added.
"There's quite a bit of rough at both ends of the pitch and it is starting to break up, it is so dry and cracked. So hopefully that keeps on happening and we can get a bit of spin and it keeps low."
England won the opening test by 74 runs and then sealed a 26-run victory in the second to secure their first series victory in Pakistan since 2000-01.
