Ashwin took 4-63 in Sri Lanka's second innings after also bagging four wickets in the first innings to take his career tally to exactly 300.
Lillee took 56 matches to reach the mark while the 31-year-old Ashwin achieved the feat in his 54th.
For India, off-spinner Ravichandra Ashwin, who became the fastest bowler to take 300 wickets in Test history, scalped eight wickets.
The off-spinner overtook former Australian speedster Dennis Lillee to set the new record.
Kohli and Ashwin have been winning matches single-handedly over the last few years.
Apart from this, the Indian team has been selected for the remainder of the series against Sri Lanka while rest of the squad remains unchanged.
As expected, the Senior National Selectors rested skipper Virat Kohli for the one-day global series against Sri Lanka and elevated his deputy in limited-overs cricket Rohit Sharma as the captain for the three-match rubber, beginning December 10 in Dharamsala.
"I really hope I can go on to double these 300 wickets, I've only played 50 Tests", declared Ashwin after the game.More news: Sacramento Kings vs Golden State Warriors: Lineups, preview & prediction 11/27/17
Beginning the fourth day 384 runs behind and needing to bat about five sessions to save the Test, Sri Lanka failed to step up to the pressure and were bundled out for 166 in just 49.3 overs.
The tourists lost seven wickets in an extended morning session and India returned after lunch to complete the formalities.
Karunaratne's overnight partner Lahiru Thirimanne played a careless shot way outside off stump to hand a simple catch to Jadeja off Yadav for 23.
"The pitch was very good to bat on and there wasn't much in it. Our batting let us down again", Chandimal said after the match. "Murali Vijay and Cheteshwar Pujara batted well too".
ODI squad: Rohit Sharma (captain), Shikhar Dhawan, Ajinkya Rahane, Shreyas Iyer, Manish Pandey, Kedar Jadhav, Dinesh Karthik, MS Dhoni (WK), Hardik Pandya, Axar Patel, Kuldeep Yadav, Yuzvendra Chahal, Jasprit Bumrah, Bhuvneshwar Kumar, Siddarth Kaul. "There's not enough time in between to prepare so we have to use the time we have now for a big tour coming up".
The first test in Kolkata was drawn.
Both the teams will now head to New Delhi, where they will face-off for the third Test starting on Saturday.