Ben Foakes, who made a key 107 and 37 on debut as the replacement wicketkeeper, was named man of the match.
England had secured a first innings lead of 139 and on Thursday an unbeaten 146 by Keaton Jennings, his first century since a ton on his 2016 debut, allowed Root to declare at 322 for six.
"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.
"We played some really good stuff", Root continued.
"It's a really hard day for us because we all know how much Rangana has done for the team and Sri Lankan cricket", said Chandimal in the post-match presentation.
'Credit goes to England, they played some outstanding cricket. But him coming into runs is a good sign for us. "We have to look at a replacement or standby player".
Kaushal Silva and Dimuth Karunaratne had denied them for more than an hour for a 51-run stand and might well have batted the entire session had they focused on defence rather being tempted into more aggressive strokes.
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Victoria Police Chief Commissioner Graham Ashton told a press conference the attack was being treated as an act of terrorism. Footage from the scene shows a man swinging a knife at two uniformed officers near a burning auto before he was shot.
And when Leach trapped Kaushal to make the breakthrough, the wickets started to fall with some regularity, his opening partner gone soon after to Moeen caught and bowled.
Another fine Moeen delivery saw Akila Dananjaya (8) edge to Stokes and then Adil Rashid (1-59) belatedly got in on the act, Dilruwan Perera (30) gone before Rangana Herath was rescued by an appeal after the on-field umpire had the retiring star out first ball.
This earned England it's biggest win against Sri Lanka in terms of runs, eclipsing a 137-run win at Lord's in 1991. Angelo Mathews scored his second fifty of the match and Kusal Mendis contributed 45 runs to the cause but the rest of the batting order could not hold up and bundled out for 250 runs.
Karunaratne enjoyed a slice of luck on 16 when he edged Moeen, but Stokes grassed an easy catch at slip.
Watch the second Test between Sri Lanka and England live on Sky Sports Cricket from 4am on Wednesday.
Jennings played out five balls on 98 before tea but the wait did not knock him off his groove and he safely strolled through for two more runs after play resumed.
"After that, the way they batted, they took the game away from us".