Ashwin finished with 4-63 in the second innings, overtaking former Australian paceman Dennis Lillee as the fastest to 300 test wickets in the process.
Lillee took 56 matches to reach the mark while the 31-year-old Ashwin achieved the feat in his 54th.
"I really hope I can go on to double these 300 wickets, I've only played 50 tests," said Ashwin, who has been working on his variations with India slated to play a number of away series next year.
"It's not easy bowling spin, it looks like you're just ambling up. But there's a lot behind it."
The tourists lost seven wickets in an extended morning session and India returned after lunch to complete the formalities.
Sri Lanka captain Dinesh Chandimal hit a defiant 61 and added 58 with Suranga Lakmal, who remained unbeaten on 31, to stretch the contest to the second session.
Paceman Umesh Yadav ended the partnership after the interval when Chandimal pulled one straight to Ashwin at fine leg.
"Before we came here, we had a gameplan," Chandimal said at the post-match presentation. "We said to the guys 'if you get a start, you have to bat long'. But they scored only 50-60 runs and got out. It's unfortunate but I'm sure they will learn."
India defeated the islanders by a 9-0 margin across formats earlier this year and after the draw in the rain-affected opening test in Kolkata, things were back to normal for the world's top-ranked test side.
After resuming on 21-1, Sri Lanka kept India's bowlers at bay for six overs before Dimuth Karunaratne was out on 18 to a freakish dismissal off left-arm spinner Ravindra Jadeja.
Lahiru Thirimanne played an irresponsible shot to a wide delivery from paceman Umesh Yadav to be caught at point for 23 in a further blow to Sri Lanka's chances of resisting India.
Ashwin dismissed Dasun Shanaka, Dilruwan Perera and Rangana Herath before lunch and then bowled Lahiru Gamage to enter the record books.
India captain Virat Kohli, who was named player of the match for his 213 in the first innings, said the comprehensive win will boost his side's confidence ahead of the tough tour of South Africa early next year.
(Reporting by Sudipto Ganguly in Mumbai; editing by Amlan Chakraborty)
