Ravindra Jadeja, who scored his maiden worldwide century during India's only innings, took the final wicket of Shannon Gabriel for four as the hosts celebrated an utterly dominant win over the former cricketing powerhouse.
However, Prithvi Shaw stitched a 206-run stand alongside Cheteshwar Pujara before Pujara fell for 86.
Kohli also came to news in this Test after two supporters entered the ground during India's batting and reached him to take selfies with the ace batsmen.
Windies captain Kraigg Brathwaite, standing in for the injured Jason Holder, was removed by Ashwin for 10 before lunch, but fellow opener Kieran Powell provided some resistance by hitting 83 from 93 balls, including eight fours and four sixes.
The margin of victory surpassed India's previous record, set when they beat Afghanistan by an innings and 262 runs at Bengaluru earlier this year.
Even before stepping on the green pitch of Rajkot, the Men in Blue were heavily touted to thrash the travelling West Indies side.
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Klopp said: "I turned to ( Chelsea boss) Maurizio Sarri and I was smiling and he looked at me like, "You enjoyed that". He said: "The problem was in the Premier League we conceded three in five or six minutes".
Resuming on 94-6, West Indies showed some fight as Roston Chase (53) and Keemo Paul (47) continued their seventh-wicket partnership from Friday evening and added 73 runs overall with some attacking batting. "Maybe it was their strategy against the spinners but it clearly didn't work this time", he said. Ajinkya Rahane contributed 41 runs. Yadav then took over.
He used his googly to send back Sunil Ambris, who stepped out without having any idea of the way the ball will turn.
Jadeja and Ashwin struck late in the session to remove Paul (15) and Devendra Bishoo (9) respectively. The pair continued this morning where they left off. Kohli in particular took a liking to Shannon Gabriel's pace, dispatching the Windies- quick for a handful of boundaries.
The match will be best remembered for the emergence of Prithvi Shaw, who announced his arrival in worldwide cricket with a classy hundred on debut, becoming the youngest from the country to do so in Tests. "Some of the other guys might be a bit disappointed but we'll crack on".
"I couldn't have imagined my test debut like this".
While the West Indies' Super50 (National One Day Meet), which started on October 3, is considered to be a parameter of selection, the schedule of the India series became an impediment towards selecting Pollard and Narine, who are playing the tournament to mend fences with the board. "Was looking to play my natural game, like I do in first-class".