Rohit Sharma's side wrapped up victory inside three days with Yashasvi Jaiswal claiming the player of the match award after the hard-hitting left hander finished on 171 in his debut test appearance as India declared on 421 for five.
That left the West Indies trailing by 271 runs and Kraigg Brathwaite's team never looked like forcing their visitors to bat again, with Ashwin continuing his bowling form from the first innings.
"Our bowling was superb. Bowling them out for 150 actually set the game for us," Rohit said after their victory.
"On that pitch, we knew we wanted to bat once, bat long."
The comprehensive defeat deepens the sense of gloom in the West Indies camp, who are smarting from their failure, for the first time, to qualify for this year's 50-overs World Cup in India.
Tagenarine Chanderpaul's was the first wicket to fall, succumbing lbw to Ravindra Jadeja with the score on eight before Ashwin secured his first victim of the second innings 14 runs later when Brathwaite was caught by Ajinkya Rahane.
Ashwin, who claimed a five-wicket haul when the hosts first went into bat, then ripped through the West Indies, with Alick Athanaze top scoring for the home side with 28 runs before becoming one of the off spinner's seven victims.
Jomel Warrican's wicket was the last to fall when he was trapped lbw as Ashwin wrapped up a comfortable victory with two days to spare.
Between them, India's spin duo of Ashwin and Jadeja claimed 17 of the 20 West Indies wickets in the match.
Brathwaite said it was difficult to make a comeback into the contest after their below-par first innings total.
"Didn't think it spun a lot early on. We just let ourselves down with the bat," he said.
"First innings total was just not good enough."
The teams will meet again in the second and final test in the series in Port of Spain from July 20 before facing off in three one-day internationals and five Twenty20 matches.