Set 205 to win, England was faltering at 70-3 but Root and Harry Brook (32) steadied the innings before the former captain guided the team home to its target as shadows lengthened on a sun-baked evening at Old Trafford.
The English finished on 205-5 to secure a fourth straight test victory this summer, after beating the West Indies 3-0 in July.
The win in Manchester was also significant in that it might signal a reset of “Bazball” — the nickname of the expansive approach favored by the leadership team of captain Ben Stokes and coach Brendon McCullum.
Root and his teammates picked their moments to attack as the run-rate fell to around 3.5 per over by the end of the second innings.
Earlier, Sri Lanka was dismissed for 326 soon after lunch, collapsing following the departure of No. 7 batter Kamindu Mendis for 113 — the third century of his four-match test career. Dinesh Chandimal was the last batter out for 79, with the final four wickets tumbling in 26 balls.
Sri Lanka added 122 runs on the fourth day after resuming its second innings on 204-6.
