Having set West Indies 269 runs to win, India hit the jackpot when Corey Collymore was caught behind for a duck off Kumble top-edging a cut, as the home team were dismissed for 219 in their second innings about 2-1/2 hours after tea.
The win gave India a 1-0 victory in the four-Test series, after the first three Tests at the Antigua Recreation Ground, the Beausejour Cricket Ground, and Warner Park all ended in draws.
It also provided India with a series victory in the Caribbean for the first time in 35 years, only their second series victory in the Caribbean in nine trips, and their first win at a ground in the Caribbean outside of Port of Spain.
Wicketkeeper/batsman Denesh Ramdin struck eight fours and one six in the top score for West Indies with an undefeated 62 from 85 balls in close to two hours.
He offered West Indies a glimmer of hope, when he added 36 for the eighth wicket with Jerome Taylor, who scored 20, and put on 39 for the ninth wicket with Pedro Collins, who made three, but there was to be no comeback story for the home side.
Ramnaresh Sarwan supported with seven boundaries in his 51 from 97 balls in three hours of batting.
The West Indies vice-captain added 70 for the fifth wicket with Dwayne Bravo, who struck six fours in 33 from 44 balls in an hour, to give India a scare.
Shantha Sreesanth, who bowled with typical vigour, made the breakthrough, when Sarwan was caught low at first slip, and Anil Kumble bowled Bravo to leave West Indies 128 for five, and effectively the match was over as a genuine contest.
Sreesanth jolted West Indies almost immediately, when he had Chris Gayle caught at second slip with the second delivery of the innings before lunch to condemn the left-handed opener to a pair of ducks for the match.
West Indies reached 18 for one at lunch, but the match swung decisively India's way with the departure of opener Daren Ganga and West Indies captain Brian Lara in the space of six overs after the interval.
Sreesanth bowled Daren Ganga for 16 in the fourth over after lunch, and two overs later, Munaf Patel scalped the big fish of batting superstar Lara lbw for 11 playing back and across to a well-pitched delivery.
Shivnarine Chanderpaul came to the wicket and spent an hour over 13 before he was adjudged lbw to Kumble to leave West Indies on 56 for four.
Bravo came to the middle and with Sarwan brought the crowd to its feet with some entertaining strokes to carry West Indies to 85 for four at tea.
After the break, both batsmen continued to bat with some enterprise and, in the fifth over after the break, Bravo hit Kumble for three successive fours, and seemed to energise Sarwan.
In the next over, the West Indies vice captain spanked two boundaries off Sreesanth, the last of which was a sweetly-timed drive through extra cover that carried him to his 50, but two balls later, the Indian fast bowler was celebrating, and West Indies were under more pressure.
Though, Ramdin and the lower order made Indian hearts flutter, there was no denying Rahul Dravid and his men earned their moment in time in the golden sunshine of the Jamaica capital.
Earlier, India were dismissed for 171 in the second innings about 45 minutes before lunch.
Dravid hit one-dozen boundaries in the top score for his side of 68 from 166 balls in a little over 4-1/4 hours to earn the Man-of-the-Match award, and ultimately, the Man-of-the-Series prize.
It was the Indian captain's second half-century of the match following his 81 in the first innings.
Collymore was the most successful West Indies bowler with five wickets for 48 runs from 24.1 overs.
It was the third time he had collected five wickets or more in a Test at this ground, and carried his tally of wickets here to 27 in four matches.
Hometown kid Taylor supported well with four for 45 from 15 overs to end with match figures of nine for 95.
India had continued from their bedtime total of 128 for six, but they were kept under pressure by disciplined West Indies bowling.
Kumble had continued to defy the Windies attack along with Dravid through the first half-hour before he was caught at third slip off Collymore for 10 to trigger a decline that saw the visitors lose their last four wickets for 30 runs in the space of 44 balls.
Dravid's 4-1/4 hours of resistance finally came to an end, when Collymore bowled him with a delivery that kept low.
Sreesanth was caught at second slip for 16 top-edging a short, rising ball on the charge off Taylor, and Harbhajan was caught low down at second slip off Collymore to give Lara his fourth catch of the innings.