England's attack made short work of the last four Pakistan batters on day five, dismissing the hosts for 200 to win by an innings and 47 runs and draw first blood in the three-match series.
Leach led the England attack with 4-30 ending Pakistan's only meaningful partnership on the fifth day when he dislodged Salman Agha for a fighting 63.
Pakistan were in danger of losing the match a day earlier when Salman and Aamer Jamal came together with the score on 82-6.
But the duo saw out the remainder of play and made England wait another 12 overs on the final day before Leach struck to open the floodgates.
Leach took a smart return catch to get Shaheen Shah Afridi for ten and then had Naseem Shah stumped for six, wrapping up Pakistan's second innings.
Last man Abrar Ahmed was unable to bat after being taken to hospital with a high fever.
Pakistan's collapse marked a quick turnaround to the match after a flat Multan stadium pitch saw 1,379 runs scored for the loss of just 17 wickets.
Brook smashed 317 and Root a record-setting 262 in England's mammoth 823-7 declared, giving the visitors a 267-run lead.
Their 454, England's highest-ever partnership for any wicket in Test cricket, gave the visitors an improbable advantage after Pakistan's impressive 556 first-innings total.
But by close of play on the fourth day on a lifeless pitch, England had posted the fourth-highest Test innings total of 823-7 and taken total control of the match with the hosts teetering at 152-6.
Along the way, Root became the highest run scorer for England in Test history.
Pakistan captain Shan Masood has now six consecutive defeats since being appointed skipper last year, including three in Australia and two at home at the hands of Bangladesh.
The hosts have gone 11 Tests without a win on home grounds, their last win against South Africa came in February 2021.
The second Test starts at the same venue from Tuesday while the third is in Rawalpindi from October 24th.