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The 211-run victory was the first away win for England in 14 Tests going back more than two years, a first on the road under Root's captaincy and only a fourth defeat for Sri Lanka at Galle in their last 14 matches at the venue.
Moeen claimed 4/71 as he and fellow spinners Jack Leach and Adil Rashid shared eight of the 10 Sri Lankan wickets.
England made 342 in the first innings thanks to a century from Ben Foakes, filling in as wicketkeeper for the injured Jonny Bairstow.
"I'm very happy with the way Mathews conducted himself throughout the game, and his commitment for running between the wickets and for the batting", he said.
"Credit goes to England, they played some outstanding cricket". We couldn't give him a really good farewell in his last game, so we apologise for that.
Fifteen without loss overnight, openers Kaushal Silva and Dimuth Karunaratne withstood the pressure from England's formidable bowling attack for the first hour. The ball spun just enough to take the edge and go to Ben Stokes at first slip, who got both hands to it but failed to grab on.
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Having charged down the track to hit Leach for a boundary on the previous delivery, Mendis tried to repeat the feat and ended up holing out to Moeen.
He had earlier appeared confident at the crease, employing the sweep shot especially well against all England´s spinners, while secure in defence.
Leach (3-60) then bowled Dinesh Chandimal for one, the Sri Lanka captain struggling to move because of a hamstring injury. Niroshan Dikwella fell the first ball after tea, caught adroitly by Stokes one-handed at slip off Ali for 16.
Even Angelo Mathews could easily have fallen for a trap that England set for him early in his innings, when he pulled a short ball from Ben Stokes to the midwicket fielder, but he earned a reprieve as he was dropped by James Anderson.
The remainder of the innings was a procession against a tired fielding unit, Jennings adding his next 46 runs at a relative canter alongside Jos Buttler (35) and Foakes (37).
England´s varied attack, with three quicks and three spinners, applied substantial pressure, with the slightly expensive Rashid the only minor disappointment.
Kavisha Dilhari, Yashoda Mendis, Udeshika Prabodini and Shashikala Siriwadana are all players who have played a fair number of T20 matches, but as to how to they fare against a team that knows how to win big tournaments and has won the World T20 once remains to be seen.