Twelve wickets fell on the opening day and India were themselves reduced to 28-2 before they recovered to post 128 without further loss by stumps at the Galle International Stadium.
Read: The final hurrah: India looking to spoil Sangakkara’s farewell series
Opener Shikhar Dhawan was unbeaten on 53 and skipper Virat Kohli was on 45 during a third-wicket stand of 100, leaving India just 56 runs away from the lead.
Dhammika Prasad trapped Lokesh Rahul leg-before for seven and Rohit Sharma fell in similar fashion to Angelo Mathews for nine, but Dhawan and Kohli stepped in to deny Sri Lanka further success.
Ashwin, playing a Test match on Sri Lankan soil for the first time, finished with six for 46 -- his 11th five-wicket haul -- as the home team were bowled out in 49.4 overs after they won the toss and elected to bat.
Leg-spinner Amit Mishra, one of the three spinners in the Indian team on an unusually dry pitch, took two wickets off successive balls to hasten the end.
Ashwin said the two early wickets taken by seamers Ishant Sharma and Varun Aaron, which reduced Sri Lanka to 15-2, made his work easier.
"Once the seamers did their job, it was very simple after that," he said. "I just wanted to get into my stride pretty strongly and start with a good rhythm."
Ashwin said he expected the pitch to play slower as the game progressed, making it tough for both batsmen and bowlers.
"The wicket was really tacky in the morning and the ball was just deviating enough," he said. "But as the game goes on, it is going to get really slow. One will have to work harder to get wickets.
"From that aspect, we need to bat really well and see where we go from here."
Mathews top-scored for Sri Lanka with 64 and Dinesh Chandimal led a charmed life to make 59, the pair sharing a valuable 79-run partnership for the sixth wicket after their team had collapsed to 60-5 before lunch.
Chandimal -- lucky to survive a dropped catch by wicket-keeper Wriddhiman Saha off Ishant in the second over after lunch when he was on five -- hit nine boundaries.
Mathews brought up his half-century with a six over long-on against veteran spinner Harbhajan Singh. He also hit six fours.
Chandimal said the early moisture in the wicket made batting difficult, but acknowledged that the Indians bowled well.
"It was not easy to bat in the morning and India used the conditions well to make it tough for us," he said. "The pitch eased out later in the day.
"We have not seen so much turn on the first day at Galle before. This sort of turn, one usually gets on the fourth or fifth day. We will try to restrict India as much as we can and obviously try and bat better in the second innings."
Kumar Sangakkara, playing the first of his last two Tests before retiring from international cricket, walked in to the bursting of firecrackers and warm applause from some 1,500 home fans.
But the left-hander, the leading run-scorer in Tests among those still playing, faced just 12 deliveries before falling in the first over after the drinks break for five.
Sangakkara punched the third ball from Ashwin straight to silly-point fielder Rahul, who gleefully held on to the sharp chance.
Mishra removed Chandimal and Tharindu Kaushal off successive deliveries, but last man Nuwan Pradeep denied the bowler a hat-trick.
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