England posed the question of whether Sri Lanka's batsmen possessed the resolve to take the game to the final day, and they fell short badly as spinners Moeen Ali and Jack Leach, playing his third test, shared seven wickets.
Moeen Ali was the pick of the England bowler, claiming 4-71 and finishing with a match haul of eight wickets.
The second match of the three-test series is in Kandy, starting on November 14, before the final contest in Colombo from November 23. "We have so much more hard work to do but to see so many talented players in the XI, or in the squad, or the different options we can go with is credit to the whole group".
In England's second innings an assured unbeaten 146 by Keaton Jennings, his first century since a ton on his 2016 debut that has silenced his critics for now at least, allowed Root to declare at 322 for six.
Beginning day four on 15-0, Sri Lanka needed a solid start to prolong their battle and overnight batsmen Dimuth Karunaratne and Kaushal Silva saw through the first hour without any damage as they put on 51 runs for the opening stand.
"Jonny, fully fit, would have played this Test match, probably would have performed well, done well with the gloves and scored runs and we wouldn't be having this conversation", insisted Prior.
"When you play Test cricket, you need to understand how people are going to get you out, even at club cricket you need to understand that", Hathurusingha said.
Ben Foakes, who made a key 107 and 37 on debut as the replacement wicketkeeper, was named man of the match. But him coming into runs is a good sign for us.
The hosts were dismissed at 203 in the first innings.
Dhananjaya de Silva, who had made a confident start to his innings, was caught at first slip by skipper Joe Root off Ben Stokes in the last over of the session for 21. He accounted for Niroshan Dickwella with the first ball after tea, Stokes making amends for his earlier drop with a sharp take, then added a soft shot from Mathews (53) to his collection.
"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.
His captaincy for the wicket of Kusal Mendis was good because after Mendis hit over the top a ball earlier, Root could have moved the fielder back - but he kept Moeen Ali up and he took the catch at mid-off.
He was followed into the pavillon by Akila Dananjaya for 8 before Herath was run out for five.
"It is very disappointing with the way we got out. getting out to the traps they set".