Fast bowler Shantha Sreesanth and leg-spinner Anil Kumble snared two wickets apiece as West Indies, leading by 219 on first innings, reached 113 for four in their second innings when stumps were drawn on the penultimate day.
Venkatsai Laxman hit an even 100 - his 10th Test hundred - as India, replying to West Indies first innings total of 581, were bowled out for 362 about half-hour past the scheduled tea break.
Corey Collymore was the pick of the West Indies' bowlers with three wickets for 63 runs from 25 overs, and Jerome Taylor snared three for 118 from 26 overs.
Dwayne Bravo collected two for 38 from 17.3 overs, and Collins captured two for 117 from 29.3 overs.
Sreesanth then had left-handed opener Chris Gayle caught down the leg-side for three in the second over of the West Indies' second innings to give India an early breakthrough.
After playing with some enterprise to hit four boundaries, West Indies vice-captain Ramnaresh Sarwan was caught at slip off Sreesanth to leave the home team on 46 for two.
West Indies captain and batting superstar Brian Lara arrived and he too, brought the crowd to its feet with a couple sixes off Kumble, but the wily leg-spinner outfoxed him with a flight delivery and had him stumped for 19.
Kumble then held a return catch to dismiss left-hander Shivnarine Chanderpaul for 11 to leave West Indies on 102 for four before Daren Ganga, unbeaten on 41, and Dwayne Bravo, not out on nine, batted through to the close.
Earlier, West Indies had tightened their grip on the match, when they captured four wickets - three to Jerome Taylor - before lunch.
Taylor pulled the rug from underneath the Indian batting in the space of six deliveries inside the first half-hour to put the skids on the visitors, after they had continued from their bedtime total of 150 for two, needing to reach 382 to avoid the follow-on.
Taylor had India captain Rahul Dravid adjudged lbw for 22 playing half forward to an off-cutter in the fast bowler's first over of the day. In his second over, Taylor struck twice in the space of three balls.
Left-hander Yuvraj Singh was caught behind for a four-ball duck edging a flat-footed drive, and Mohammed Kaif was dubiously adjudged lbw for a two-ball duck, as television replays suggested the ball may have been drifting down the leg-side.
Laxman remained unflappable and reached his 50 when he drove Taylor into wide mid-on for a single and offered not even a semblance of a chance.
He held the Indian innings together before the interval and spent an hour and 45 minutes with Mahendra Dhoni trying to repair the early damage with a stand of 61 for the sixth wicket.
But Collymore had Dhoni adjudged lbw for 29 when the batsman played back and across to a well-pitched delivery that moved back from outside the off-stump. After lunch, Laxman and Anil Kumble resisted to add 77 for the seventh wicket with little trouble.
During this period, Laxman completed his hundred, when he steered a delivery to third man for a single to reach his hundred, but he was sadly dismissed with the first ball of the next over.
Collins made the breakthrough for West Indies, when he had Laxman caught behind playing on the back-foot and edging a ball angled across him after he struck 15 boundaries from 231 balls in just over six hours.
Three overs later, Collymore had Anil Kumble caught at deep fine leg for 43 top-edging a hook at a short, rising ball. The Indian leg-spin bowler smote eight fours from 61 balls in 96 minutes.
Collins then had Shantha Sreesanth caught at second slip for a second-ball duck top-edging an upper cut, but Patel came to the crease and upset the West Indies' plans before slicing Bravo to gully.
India were 315 for nine about 40 minutes before the scheduled tea break, but their last wicket pair of Harbhajan Singh, who was not out on 38, and Munaf Patel then spent an hour and 10 minutes over a stand of 47 that beefed up the total.
The partnership cut into the West Indies' lead, and was finally broken, when Bravo had Patel caught at gully for 13.
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 series concludes at Kingston's Sabina Park from June 30 to July 4. Warner Park is staging its first Test. It has become the ninth Test venue in the West Indies and the 94th in the World.
Scoreboard at close on the fourth day of the third Test between West Indies and India at Warner Park here on Sunday:
West Indies 1st Innings: 581 (D. Ganga 135, R. Sarwan 116; Harbhajan Singh 5-147)
India 1st innings (overnight 150 for two)
Fall of wickets: 1-61, 2-124, 3-157, 4-159, 5-159, 6-220, 7-297, 8-311, 9-315
West Indies 2nd innings:
To bat: M Samuels, D Ramdin, P Collins, J Taylor, C Collymore
Fall of wickets: 1-3, 2-46, 3-81, 4-102
Umpires: R. Koertzen, B. Jerling
TV Replays: C. Mack
Reserve: L. Kelly
Match Referee: J. Crowe