Root's 124th in England's second round also played a key role in winning 57 victories and secured him the best man award.
Celebrating his first win away from England since winning South Africa in 2015-16 and his first triumph as captain, Root sees England – currently ranked third in the world – challenging India to first place on the test team .
"It was a brilliant test game," said Root, whose team had to fight back twice after their first-round scouts gave cheap wickets.
"It has dropped and flowed everywhere, as against India in the summer, and it is nice to see another game coming on our way.
"We said we would go out and play cricket, and we will be brave. It is easy to talk about a good game, but we support it," said Root.
"We have to keep improving, the team has grown in the last 18 months, but we are not the finished article, we want to reach the world number one and that means acting in all conditions around the world."
Sri Lanka needed a 301 in their second inning to record a frustrated win, but only managed 243 at the Pallekele field.
They started the day needing 75 more races and three wickets in hand had an outside chance.
After Niroshan Dickwella was caught by Ben Stokes in Moeen Ali for 35 – the first shutter in the morning – the hopes of Sri Lanka were almost erased.
Moeen then beat No. 10 batter Suranga Lakmal, the captain of Sri Lanka, on the next ball of the court for a strong call to the lbw who was refused. He released Lakmal his next installment.
Pushpakumara lasted only eight balls before kicking the ball to Leach and the game is over.
Lakmal insisted that his team had been "unlucky." But England's triumph followed a landslide victory of 211 races in the first test in Galle, and was established by masterful performances of Leach, Ali and Rashid.
Spinners dominated on both sides as Sri Lankan Dilruwan Perera took seven wickets in the game and Akila Dananjaya – who is now set aside for an investigation into his bowling action – eight.
"It's good to have three spinners on the team, we all know our roles and we help each other, it's good to make history and we hope we can continue with this now," Rashid said.
Leach, just playing his third test, said the game was "unbelievable."
Root insisted that England will not facilitate the third and final test, which starts in Colombo on Friday.
He faces some difficult selection choices after goalkeeper Ben Foakes has shone again with a baton – reaching a crucial invincibility of 65 in the second round – just as Jonny Bairstow raises recall claims after injury.
Sri Lanka will try to strengthen its reconstruction, said coach Chandika Hathurusingha.
"We're looking at the bigger picture," he said.
"This is the first series we lost this year, we won 2-0 against South Africa, we had very close games in both games, we had the opportunity to win both games at times.
"England played better cricket than we do, we have to accept that, we have to learn from those mistakes," he added.
"The ability is still there – that's what I'm betting on. That's the hope I have with this team."