Root dug in after England had made a shaky start to chasing a modest victory target of 205 on Saturday, reaching his match-winning unbeaten half-century off 108 balls.
But Sri Lanka made a real game of it after another fine century from Kamindu Mendis forced England to have to labour for their victory.
Kamindu Mendis hit a superb hundred for Sri Lanka - but it proved in vain. (AP PHOTO)
"We had to work hard for that over the four days but we bowled really well throughout, Jamie Smith batted so well and then Joe put on a clinic for us at the end," England captain Ollie Pope said.
"I thought Sri Lanka played really well and pushed us all the way."
England lost opener Ben Duckett early when he edged to wicketkeeper Kusal Mendis for 11 and Pope soon followed him back to the pavilion, caught at slip for six.
Dan Lawrence (34) was next to go, trapped lbw by Milan Rathnayake. Harry Brook (32) shared a 49-run partnership with Root before being caught and bowled by Prabath Jayasuriya, leaving England wobbling at 4-119.
Smith, who scored his maiden Test century in the first innings, matched Root's patience as England went an hour without a boundary and the hosts were firmly back in control before Smith was bowled by Asitha Fernando for 39.
Root, who ended unbeaten on 62, hit only his second four from the last ball of the match to give England a 1-0 lead in the three-Test series.
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Sri Lanka captain Dhananjaya de Silva shrugged: "We had our plans but in the first innings but we couldn't execute them. Our mistakes were made in the first innings and that cost us the match."
Sri Lanka had started the day at 6-204 and Mendis alongside Dinesh Chandimal batted through the rain-hit first session to forge a 117-run partnership for the seventh wicket. It got the touring side past the 300-run mark.
Kamindu (113) scored his third Test century as he kept the scoreboard ticking with a combination of sublime shot-making and tireless running between the wickets.
The middle-order batter has fast become one of Sri Lanka's most consistent Test players, scoring three centuries and two half-centuries in the eight innings he has played in the longest format of the game.
But Kamindu nicked Gus Atkinson to Root soon after lunch as England made the most of the new ball, despite fast bowler Mark Wood being ruled out for the day with a thigh injury.
Matthew Potts dismissed Chandimal, who scored 79 despite playing through pain after suffering a blow to his thumb, to end Sri Lanka's innings on 326, with the visitors adding only 19 runs for the last three wickets.
The second Test starts at Lord's on Thursday.
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