- Tag results for Joe Root
England got off to a shaky start in Galle, losing five wickets for just 103 runs, with debut opener Rory Burns back in the pavilion for nine, Moeen Ali out first ball and Ben Stokes gone for seven.
England's varied attack, which features three seamers and three spinners, probed persistently through the morning as captain Root employed attacking field placements.
Eoin Morgan hit 11 fours and two sixes in his 91-ball innings that formed the core of the England score after Sri Lanka won the toss and put the tourists in.
Warne, who is promoting his autobiography, said Root is England's best player but would be disappointed with his conversion rate when it came to hundreds.
Jimmy Anderson overtook Australia's Glenn McGrath to become the most successful fast bowler in Test history, with 564 wickets.
Joe Root's England side are now just one point behind South Africa and Australia, who are both on 106 points with the former marginally ahead on decimal points.
The day certainly belonged to Cook, who scored his 33rd hundred his last appearance to join a select band of players having scored a hundred on debut and final Test.
Cook's 181-run third-wicket partnership with his successor as captain, Joe Root, who is unbeaten on 92, has taken England's lead to 283.
Buttler and Stuart Broad frustrated India with a ninth-wicket stand of 98 as, yet again this series, England bounced back with the bat after a top-order collapse.
Cook, who walked out to a guard of honour from the India team, was soon into his stride with two trademark boundaries off successive balls from fast bowler Jasprit Bumrah.
Root said that Cook had informed him of his retirement decision in Southampton, and that this wouldn't be a distraction ahead of or during the fifth Test.
Bairstow had broken a finger keeping during India's 203-run win in the third Test at Trent Bridge.
Root said he was confident of a win with so many bowling options at his disposal as England's potent attack dismissed world No 1 India for 273 and 184 in the two innings to secure the win.
Moeen Ali started the fourth test as England's second spinner. By the end of it, he was England's No. 1 bowler.
Buttler scored 69 and put on vital lower-order partnerships with Ben Stokes and Sam Curran as England recovered from 122 for five to finish at 260 for eight at stumps on day 4.