The visitors then put themselves in a strong position to complete a remarkable victory in Multan by reducing Pakistan to 6-152 at stumps on day four
Match Report:
ScorecardBrook eclipses legends in England's astonishing 823
Harry Brook has eclipsed some of English cricket's most prestigious names with his majestic triple century against Pakistan, but admitted his father's family record was the one he really wanted.
Brook smashed a sensational 317 on day four of the first Test in Multan on Thursday, powering the tourists to the fourth-highest total in history - an outlandish 7-823 declared.
The touring attack then set themselves up for a final victory push by reducing Pakistan to 6-152 by stumps, still 115 behind.
Until Brook, no Englishman had made a treble since Graham Gooch's 333 against India in 1990 and, in doing so, the Yorkshireman bettered the best days of a host of illustrious names from the past and present, including John Edrich (310), Alastair Cook (294), Denis Compton (278) and Ben Stokes (258).
Even his mentor turned teammate Joe Root has never reached 300, despite making a record 35 hundreds and doubling up for the sixth time on Thursday with a career-best 262.
But the number Brook was keenest to cross off was 210, the score his father David Brook made for Burley Cricket Club against Woodhouse in a league game all the way back in 2001.
"I knew about Gooch but I didn't know about the other ones. I just wanted to get past my dad's highest score!" he said with a satisfied smile.
"I've said that before and I was pretty happy when I got past his score to be honest. It's not quite sunk in, really.
"I'm still absolutely knackered. I'm sure I will sleep well tonight and reflect on it later.
"I've never heard of a total like that before; me and Rooty just tried to cash in on what was a good pitch. It was a tiring day... I'm lost for words."
Root, who put on an unprecedented 454 alongside Brook, suggested his new mark as England's top run-scorer might one day go to his fellow Yorkshireman.
"Harry has got such a complete game. To see him go and get a massive score is awesome. I'm not surprised at all in him going on and doing something special like that, but I don't think it'll be the last time we see him with a monster score by his name," said Root.
"I knew how good he was going to be, it was just when he was going to figure it out. The pace he has matured into the player he is is probably the one thing which has shocked me."
Brook's innings took him to fifth on England's all-time list while his monster stand with Root easily eclipsed the previous best of 411 between Peter May and Colin Cowdrey, against the West Indies in 1957.
Only three pairings globally have ever put on more than the Yorkshire duo and they leapfrogged even Sir Donald Bradman's highest partnership (451 with Bill Ponsford versus England at The Oval in 1934) by three runs before giving way.
England, who conceded 556 in the first innings, somehow flipped that into a lead of 267, but it became even better as Pakistani wickets began to tumble.
England's merciless scoring appeared to break the home side's spirit - the dejected looking bowlers, lethargic ground fielding, the truly terrible catching - but Pakistan's top-order implosion confirmed they had lost heart for the fight.
They lost opener Abdullah Shafique to the very first ball of their second innings, thanks to fine skill from Chris Woakes, nipping in and sending off stump spiralling.
First-innings centurion Shan Masood then survived two drops before chipping Gus Atkinson to short midwicket for 11 and the Surrey seamer had Babar Azam caught behind moments later.
Debutant Brydon Carse then took out Saim Ayub with his loosener, well caught by the retreating Ben Duckett at mid-off, and bowled off-form Mohammad Rizwan.
After just 13 wickets had fallen in the first 10 sessions of the match, Pakistan conspired to lose five in 12.1 overs. Jack Leach made it six when he had Saud Shakeel caught behind but a four-day win slipped through Shoaib Bashir's fingers as he dropped Aamer Jamal at fine-leg to deny the impressive Carse.
Earlier, Yorkshiremen Root and Brook put on 166 in 29 overs. Root was badly dropped at midwicket on 186 by Babar, and Jamie Smith was later put down twice in a late cameo as heads dropped.
Root reached 200 for the sixth time in his career and accelerated past 250 with a reverse scoop. Brook ticked off his Test best of 186, his first-class best of 194 and made his double-ton in 245 balls.