Chasing 392 for victory, India reached 298 for four in their second innings, when the two sides decided it was pointless to continue on the last day with three overs remaining.
The result means the four-Test series is still level at 0-0, after the first two Tests at the Antigua Recreation Ground and the Beausejour Cricket Ground ended in tense draws. The final Test begins on Friday at Kingston's Sabina Park.
Four Indian batsmen - Virender Sehwag, Venkatsai Laxman, skipper Rahul Dravid, and opener Wasim Jaffer - all completed half-centuries to keep the visitors in the hunt for most of the day.
Dravid struck four fours and one six in the top score of 68 from 131 balls in 2-3/4 hours and has drawn level on the Test aggregate list with England's Graham Gooch with 8,900 runs.
The Indian captain added 100 for the third wicket at the height of the chase with Laxman, whose 63 contained 10 fours from 106 balls in two hours and 20 minutes.
Sehwag had hit eight fours in 65 from 75 balls in 1-3/4 hours and Jaffer collected five boundaries in 54 from 118 balls in close to three hours to lay the platform for the Indian chase.
Sehwag and Jaffer added 109 for the first wicket to excite the chase buds of the Indians, but the game failed to be a genuine contest when Mahendra Dhoni was caught at extra cover for 20 off Jerome Taylor about an hour before the end to leave India on 273 for four.
West Indies had declared their second innings on 172 for six about 35 minutes after they continued from their bedtime total of 113 for four.
Daren Ganga, who was named man-of-the-match, struck eight boundaries from 74 balls in 1-3/4 hours in an undefeated top score of 66 for the home team that followed his Test-best 135 in the first innings.
Anil Kumble was India's most successful bowler with three wickets for 60 runs from 12 overs, and Shantha Sreesanth collected two for 19 from six overs.
Jaffer and Sehwag then emerged from a sedate start on the hard, true easy-paced pitch and gradually gained control with languid strokeplay to carry India to 109 without loss at lunch.
Fortune smiled on Sehwag twice, after he scored two to reach 4,000 runs in Tests.
On five, Chris Gayle dropped him at first slip off Jerome Taylor in the fourth over of the chase, and in the final 10 minutes before lunch, Ramnaresh Sarwan dropped him on 56 at straight extra cover off Dwayne Bravo.
Sehwag had survived a torrid new ball spell in which he was peppered by a succession of short-pitched deliveries on the body, where he is vulnerable, from Pedro Collins and Taylor, and reached his 50 with a straight drive for a boundary off Bravo.
After the interval, West Indies had immediate success, when Sehwag was adjudged lbw playing back and across to a well-pitched first delivery from Corey Collymore.
Jaffer was then caught at first slip off Pedro Collins with the drinks break in reach to leave India on 143 for two.
Laxman and Dravid, India's two most celebrated batsmen, then continued the visitors' hunt for runs - and ultimately victory - and took them to 200 for two at tea.
Laxman had driven Marlon Samuels to mid-on for a single to reach his 50, but he was dismissed about 25 minutes after the break, when he top-edged Collins and was caught at second slip.
India still needed 149 runs and a minimum of 25 overs were still available, when Mahendra Dhoni came to the crease and dispatched his first ball - from Collins - over long-off for a six.
Collins, who suffered the indignity of a second six at the hands of Dhoni later in his innings, and Collymore tightened up, and the pressure continued to mount on the Indians for quick runs.
Dhoni eventually cracked and drove Taylor ferociously to Gayle to deflate India's chances of a late charge for glory.
Earlier, West Indies predictably chose to pursue quick scoring instead of caution, and suffered an early setback, when Bravo was caught at long-on for nine off Kumble in the first over of the morning.
Ganga then swept Harbhajan Singh through backward square leg for two to reach his 50 and got things moving in a brief, but exciting partnership of 32 with Marlon Samuels.
Harbhajan made the breakthrough, when he deceived Samuels in flight and had him stumped for 20 that included three well-placed, leg-side boundaries before the declaration came.