Sri Lanka end with 67 for 3 chasing a target of 386 on day four
Resuming on their overnight score of 47 for 3, Kohli and his team managed 274 as the middle order batsmen chipped in with crucial runs.
Rohit Sharma and Ashwin hit fine half centuries while Stuart Binny missed out by a run.
The Binny-Sharma partnership for the fifth wicket added 54 runs; later Ashwin combined with Mishra to post 55 for the eighth wicket partnership and added another 35 with Umesh Yaadav for the ensuing wicket.
Rohit Sharma scored 50, Ashwin played a handy knock of 58, and Binny scored 49.
For Sri Lanka, Nuwan Pradeep picked up 4 wickets for 62 runs while Dhammika Prasad gave away 69 runs for his 4 scalps.
Setting out for the chase, the hosts lost Upal Tharanga early for naught.
Karunaratne was the next to follow as Umesh sent him packing for a duck.
Dinesh Chandimal played an enterprising knock but it was cut short by Ishant who struck for the second time to dent the Lankans.
Skipper Angelo Mathews (22) and Kaushal Silva (24) played with caution to solidify their innings.
The duo will resume batting when play starts on the fifth day.
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