Chasing an imposing 346-run target on the fifth and final day of the match, West Indies collapsed for 108 in 47.3 overs during the post-lunch session.
India had won first Test by an innings and 92 runs at Antigua on July 24 and drew the second Test at Kingston in Jamaica on August 4.
The fourth Test will be played at Port of Spain from August 18-22.
The visitors, who had a 128-run lead from the first innings, declared their second innings after 48 overs in the first hour of the morning session.
Darren Bravo was the only batsman in the West Indies' second innings who put up some resistance with a fighting 59. But he quickly ran out partners and was eventually dismissed by Indian pacer Mohammed Shami which left the West Indies struggling at 95/8.
Bravo's 100-ball stay in the middle was studded with one six and five boundaries.
Though Bravo and Marlon Samuels tried to defy the Indian bowling attack and added 29 runs for the third wicket partnership, the latter was first to go when his off-stump cart-wheeled by a beauty from Ishant Sharma.
Shami was the pick of the Indian bowlers with three wickets, while Ishant and Ravindra Jadeja shared two each. Bhuveneshwar Kumar and Ravichandran Ashwhin bagged one wicket each.
Roston Chase and Jermaine Blackwood did not last long as they got out cheaply under pressure. While Bravo survived at one-end, keeper Shane Dowrich (5) and captain Jason Holder (1) exited in the 38th and 40th overs.
A brilliant throw by Aswhin ran out Holder (1).
Tailenders Aljarri Joseph (0) and Shannon Gabriel (11) were packed off in a span of final overs while Cummins remained not out with 2 runs.
Walking into the middle to draw the match or avert a defeat again, Windies openers Kraig Brathwaite and Leon Johnson found the going tough, as Shami and Kumar swung the ball both ways to unsettle them.
As planned, Shami made Johnson pop an easy catch to Rohit Sharma at short leg for a duck in the fourth over, while Kumar trapped Brathwaite leg before in the following over for four in 20 balls to reduce the hosts to 4/2 in five overs.
Earlier, resuming the innings from the overnight score of 157/3 in 39 overs on a sunny but humid day, India lost four wickets for just 60 runs in nine overs to an attacking bowling, especially by Cummins.
Sharma, who scored a breezy 41 in 59 balls with three mighty sixes and a boundary on the fourth day (Friday), was the first to go when a straight delivery from Cummins trapped him leg before for the same score.
Though Indian wicketkeeper Wriddhiman Saha (14 in 14 balls) and Ravindra Jadeja (16 in 13 balls) made useful contributions to the score and helped Rahane to rotate the strike with quick singles and twos, both fell to Cummins who was in great form.
Brief scores: India 353 & 217/7 (Ajinkya Rahane 78 not out; Roston Chase 1/41) vs West Indies 225 & 108 (Darren Bravo 59; Mohammed Shami 3/15)
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