The Indians successfully defended a victory target of 326 when they dismissed West Indies for 262 in their second innings on the fourth day at Sabina Park.
Suresh Raina formalised the result when he bowled Devendra Bishoo behind his back for 26 about 35 minutes after lunch.
The victory gave India a 1-0 lead in the three-Test series, with the second Test set to start on Tuesday at Kensington Oval in Barbados.
The main architects of the visitors' final push for victory, Praveen Kumar finished with three for 42 from 16 overs, Ishant Sharma ended with three for 81 from 17 overs and Amit Mishra captured two for 62 from 13 overs.
India's bowlers were again virtuous with their line and length, and none of the remaining West Indies batsmen showed the kind of resolve to resist, after the hosts resumed on 131 for three.
Before lunch, six wickets -- two apiece for Kumar and Mishra -- put India on the verge, sinking West Indies to 226 for nine.
Kumar removed the overnight pair of Darren Bravo, bowled behind his back for 41, and Shivnarine Chanderpaul, caught at extra cover for 30, inside the first half hour.
Harbhajan Singh then spoiled Carlton Baugh Jr's 29th birthday celebrations, when he had him caught at leg-slip for a duck, leaving West Indies 150 for six.
India's champion off-spinner was, however, quickly removed from the attack, when West Indies captain Darren Sammy launched him for three successive sixes over long-on into the Great Northern Stand.
But Mishra cut short Sammy's fun when he had him caught at extra cover for 25 and inflicted more agony on the hosts when he had Brendan Nash lbw for nine attempting an ugly slog, leaving the hosts reeling on 188 for eight.
India again met some resistance from the West Indies tail, with Ravi Rampaul making a strong case for a promotion in the home team's batting order, stroking 34 before he was caught behind, gloving a snorter from Sharma.
The final Test between the two sides starts on Wednesday, July 6 at Windsor Park in Dominica.
