Excellent pace bowling, with Chris Woakes and Mark Wood taking three wickets each, left England chasing a modest target of 252 after the Pakistan innings ended with one ball left, which was crossed with 15 balls remaining.
Pakistan had collapsed to 2 for 3 before Sarfraz Ahmed's 105 and Imad Wasim's 63 not out lifted it. The pair put on 77 for the sixth wicket.
After a couple of early stumbles in its reply, a third-wicket partnership of 112 between Root and Eoin Morgan (68) put England on course for the win, with Morgan's knock ending a run of 23 international innings without a half-century.
England's chase suffered a setback early on when Jason Roy, who made a dashing 65 in the first game, fell for a second-ball duck as he was bowled by Mohammad Amir.
Root steadied England with two fours in as many balls off Hasan Ali but Alex Hales's lean run continued when he was bowled for 14 trying to slog-sweep Wasim. That left England at 35 for 2 but Root and Morgan took control of the game, the captain hitting nine fours before he too was bowled by Wasim.
Ben Stokes made a brisk 42 before Root, who hit just five fours in his patient knock, holed out off Wahab Riaz with England in sight of the target at 240 for 6.
Woakes ended the match with enough to spare when he cut Yasir Shah for four.
Pakistan, which won the toss and batted on a cloudy morning, lost Sami Aslam after his attempted pull off Woakes was held down the legside by Jos Buttler. Wood then dismissed Sharjeel Khan for nought with an express delivery that cut sharply in to knock over off stump. Azhar Ali also fell for a duck when caught behind off Woakes.
Babar Azam (30) struck several boundaries before he played on to Liam Plunkett, leaving Pakistan at 66 for 4.
But Sarfraz frustrated England with a mixture of orthodox and improvised shots.
Shoaib Malik (28) struck a straight six against Moeen Ali in a fifth-wicket partnership of 59 before Wood's return saw him caught behind.
Sarfraz, whose lone previous ODI hundred was against Ireland at last year's ICC Cricket World Cup, completed a 124-ball century when he glanced Plunkett for his sixth four. He then successfully reviewed Plunkett's lbw appeal before slog-sweeping Moeen to Hales in the deep.
Wasim struck Plunkett for three boundaries in as many balls – an uppercut six followed by a straight drive and a cover drive – to go to a 57-ball fifty. Their efforts, in the end, wasn’t enough.
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