Herath, 40, thus retired after getting run out in the 86th over.
Coverage: Ball-by-ball Test Match Special commentary on BBC Radio 5 live sports extra, Radio 4 LW, online, tablets, mobiles and BBC Sport app.
"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.
"It's been a privilege and honour to play for my country, because in Sri Lanka we have 22 million people, so very few get the opportunity to play for Sri Lanka, so that's a remarkable achievement and honour for any player", he added.
Overnight batsmen Dimuth Karunaratne and Kaushal Silva saw through the first hour without any damage even if they had a little luck when Karunaratne, on 16, edged Moeen Ali but Ben Stokes grassed the easy catch at slip.
But soon after the morning drinks break, Leach broke though when he had Silva out leg before wicket for 30.
Karunaratne had been given an early life by Ben Stokes at slip, England's best catcher grassing one off Moeen Ali, but did not cash in.
Thisara Perera and Rangana Herath, who was playing his final global fixture, took two wickets each.
The win helped England claimed their first away Test victory in 13 matches.
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.
Sri Lanka failed to give the retiring Rangana Herath a winning farewell but the cricketer made sure that England earned the final wicket to seal the last game of his career.
Jennings looked reasonably confident against the spinners, often playing the sweep shot - both traditional and reverse - to negate them.
Home skipper Dinesh Chandimal is unable to bat any higher than number seven because of a groin injury that kept him off the field for all of Thursday, weakening Sri Lanka's batting order. Angelo Mathews was the top scorer with 52 runs. The two put together 107 runs for the fourth wicket with Stokes taking spinner Akila Dhananjaya for two sixes and a four in two overs.
England's varied attack, with three quicks and three spinners, applied substantial pressure, with the slightly expensive Adil Rashid the only disappointment so far.