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.
Chandika Hathurusingha, Sri Lanka's coach, said his side were playing catch-up since lunch on day one when England turned the match around having been five down for 103. Mathews scored a half-century in the first innings, too.
Set a gargantuan 462 for victory, the hosts ended the morning session on 98 for 3, after their openers had earlier batted out the first hour of the fourth day.
Less than three overs later Karunaratne decided he would take the aerial route against Moeen, an ill-judged scheme that saw him club a firm return catch back at the off-spinner.
But both fell after attacking shots, with Kaushal leg before trying to sweep Leach for 30 and Karunaratne caught and bowled by Ali for 26 and skipped down the pitch for an attempted heave.
Sri Lanka lost Kusal Mendis (45) and captain Dinesh Chandimal (1) in the afternoon session.
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England captain Joe Root (three) also had a deja vu moment when Herath dismissed him for the second time in the match. England hadn't won four previous tests here.
Stokes said before the day's play that he expected Moeen, Leach and Adil Rashid to do the most damage, with the ball spinning more quickly off the pitch as the match has progressed.
England's players did a lap of honor to the appreciation of more than 6,000 of its fans at the game. Suranga Lakmal led Sri Lanka during Chandimal's absences in Galle.
England laid traps and Sri Lanka's batsmen sometimes fell into them with such willingness, it was as if they were "school kids".
The 25-year-old became just the 20th Englishman in 141 years of Test cricket to score a hundred on debut and only the second keeper after Prior, who made an unbeaten 126 against the West Indies at Lord's in 2007.
We wondered about them playing against Sri Lanka's spinners, we wondered about the pitch - whether it would be as hard as it usually is here - and we also had concerns about the weather. "For a guy to come in on his debut and play an innings of that substance is exactly what we're after, for him to stand up and deliver", Root said.
Herath retired from worldwide cricket as one of the most successful Lankan bowlers in Tests after Muttiah Muralitharan who scalped 800 wickets.
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