The visitors slipped from 109-0 to 131-7 before an unbeaten 100-run eight-wicket partnership between Mahendra Singh Dhoni and Bhuvneshwar Kumar pulled the team through.
Rohit Sharma got India off to a flier but Sri Lanka struck with wickets of Shikhar Dhawan and Virat Kohli early in the chase.
What time does India vs Sri Lanka 3rd ODI match start?
Indian player MS Dhoni scored 67 runs and remained not out till the end. Rahul was bamboozled by Sri Lanka's new mystery spinner Akila Dananjaya as he was drawn forward on the front foot into a drive.
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Kapugedara leads Sri Lanka for the first time as he stands in for Upul Tharanga who was suspended for two games by the International Cricket Council for maintaining a slow over rate in the last game.
The Men in Blue haven't played much ODI cricket here, and Thursday's game which they won by three wickets was only their second at this venue, having played previously in 2012.
With Lahiru Thirimanne threatening to cut loose later, Virat Kohli turned to Bumrah again, and he delivered a flawless slower ball which the batsman chipped straight to midwicket.
India were 210 for four in 44 overs in pursuit of Sri Lanka's total of 217 for nine in 50 overs when the frustrated crowd rained bottles down the ground from either side of the grass beds.
With opener Danushka Gunathilaka also injured and out of the team, Sri Lanka has called up test captain Dinesh Chandimal and batsman Lahiru Thirimanne, who has not played for the national team in more than a year.
PV Sindhu claims silver after gruelling, epic final
Commenting about her next encounter with Chen, Sindhu said: "She is a good player and she is playing well. Sun Yu tried to stop the juggernaut but could not do much as Sindhu raced to win the opening game 21-14.
For Sri Lanka, spinner Akila Dananjaya scalped two wickets while pacers Lasith Malinga and Vishwa Fernando took one wicket each.
The Indian skipper wants his ODI bowlers to look for wickets and Bumrah has set an example.
Lokesh Rahul, M.S. Dhoni (wk), Kedar Jadhav, Hardik Pandya, Axar Patel, Bhuvneshwar Kumar, Jasprit Bumrah, Yuzvendra Chahal.
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'Triple talaq is everything. Court's decision can not overrule our faith'
Muslim men are giving divorce to their wives by following triple Talaq practice. Dignity is guaranteed to all under the constitution.