At stumps, India were 80-0, only 70 runs behind with Yashasvi Jaiswal 40 not out and captain Rohit Sharma 30 not out.
Returning to the India team after missing the World Test Championship final in June, Ashwin took 5-60 from 24.3 overs, including the wicket of Tagenarine Chanderpaul (12).
In doing so, he became the first Indian and fifth player to dismiss a father and son in Test cricket: Ashwin dismissed Shivnarine Chanderpaul - father of Tagenarine - in his debut Test in 2011.
Mitchell Starc, Ian Botham, Wasim Akram and South Africa spinner Simon Harmer are the only other players to have dismissed a father and son in Tests: Starc and Harmer also achieved the feat against the Chanderpauls.
Ashwin also passed 700 wickets across all international formats.
"I thought it was a really good performance," Ashwin said. "There was some moisture in the pitch in the first session. It got slower and started to spin a little bit. I enjoyed the spell in the first session and in the second I had to work a little bit more."
Bowling as first change, Ashwin dismissed both openers in the morning session as the West Indies went to lunch at 4-68.
He bowled Chanderpaul with a doozy of delivery, angled in from wide on the crease which pitched on middle and straightened past the outside edge to hit off.
Kraigg Brathwaite had struggled to 20 in just over an hour when he was caught by Sharma at cover from Ashwin. The batsman tried to hit a fuller ball over midwicket but missed the length and sliced it into the off side.
The West Indies had slumped to 8-137 by tea and Ashwin had added the wickets of Alzarri Joseph and Alick Athanaze near the break.
Joseph misread the carom ball which he thought was turning from off. Instead it turned from leg and his attempt to strike the ball over the leg side resulted in a leading edge which carried to Jaydev Unadkat at point.
Athanaze top-scored for the West Indies with 47, compiled from 99 balls in 135 minutes and including six fours and a six. He looked well on the way to a half century on debut when he was undone by a shorter, faster ball from Ashwin which he sought to pull but skied to Shardul Thakur.
Ashwin returned to end the innings with the wicket of Jomel Warrican and to complete his fifth five-wicket bag against the West Indies, his fourth in the Caribbean. Rahkeem Cornwall was left 19 not out.
The India batsmen looked comfortable on a slowing pitch before stumps, bringing up their 50 partnership from 74 balls.
The match begins a new World Test Championship cycle.
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