Chasing a modest 218 for victory from their allocation of 50 overs, West Indies hit the jackpot off the last ball of the 44th over, bowled by Suresh Raina, which Wavell Hinds carted to the deep mid-wicket boundary for four.
The victory gave West Indies an unbeatable 3-1 lead in the five-match series which concludes on Sunday at the same venue, and was their first major series win, since unexpectedly clinching the ICC Champions Trophy two years ago under Lara.
It is also the first series victory for West Indies in a bilateral series against opponents rated higher than them on the ICC ODI Rankings, since they also prevailed over India 4-3 on the sub-continent four years ago.
Lara, a struggler for most of the series, showed a timely return to form to top-score for his side with 69 from 95 balls that contained eight fours and one six.
He gained admirable support from Bravo, whose undefeated 61 from 62 balls that included three fours and one six, helped to earn him the man-of-the-match award.
Lara and Bravo added 91 for the fourth wicket before the West Indies captain was caught at deep mid-off in the 40th over off Ramesh Powar, whose two wickets for 56 runs from nine overs made him the most successful Indian bowler.
chad completed half-centuries of their own, as India were restricted to 217 for seven from their allocation of 50 overs.
Kaif hit five boundaries in the top score of 62 from 84 balls, and Yuvraj collected the same number of fours in 52 from 78 balls. Dwayne Bravo snared three wickets for 32 runs from five overs to be the most successful West Indies bowler.
cbludgeoned India's new ball pair of Irfan Pathan and Shantha Sreesanth to give West Indies a positive start on his way to 46 off 70 balls that included seven fours and one six.
Gayle reserved special treatment for Sreesanth and swung him high over mid-wicket for a six that flew straight out of the ground.
Pathan made the breakthrough, when he had Marlon Samuels lbw for nine in the fifth over, and Ajit Agarkar then sent jitters into the West Indies camp, when he had the in-form Ramnaresh Sarwan caught at square leg for six trying a pull to leave the home team 49 for two in the ninth over.
Lara entered to rapturous applause from the crowd in the packed stadium and he blossomed to stabilise the innings in a stand of 42 for the third wicket before Gayle was caught at long-off off Powar in the 23rd over and Bravo arrived to carry Windies home.
Earlier, Kaif, similarly, came to the wicket with India on 47 for three in the 16th over and with Yuvraj rescued the visitors from their early instability.
Kaif and Yuvraj have shown in the past that they enjoy batting together, and gave further proof with a stand of 80 for the fourth wicket.
Yuvraj, looking to bury the demons of the second ODI at Kingston, where he was dismissed with two balls to spare to hand West Indies victory, played with his customary flair.
A straight drive off Corey Collymore for a boundary was his most memorable stroke, and he reached his 50 when he drove Smith to mid-on for a single, but he was soon dismissed, caught behind trying to steer Edwards to third man in the 32nd over.
Mahendra Dhoni, whose 46 not out off 58 balls contained half-dozen fours, arrived and continued to consolidate India's position with Kaif in a stand of 61 for the fifth wicket.
Bravo bowled Kaif in the 47th over with 21 balls remaining and India on 188 for five, leaving just enough time for Dhoni to play a few enterprising strokes and get his side over the 200-run hump.
India had suffered two early setbacks.
In the fourth over, Virender Sehwag was caught at cover for 11 off Ian Bradshaw, who also had Suresh Raina caught at slip for seven top-edging a cut in the 10th over to leave the visitors on 28 for two.
India were just beginning to make progress, when Corey Collymore struck in the 16th over to have skipper Rahul Dravid caught low down at second slip for 15 to leave the visitors 47 for three, paving the way for Kaif and Yuvraj to turn things around.