Jaffer scored 60, as India, replying to West Indies first innings total of 581, reached 150 for two when stumps were drawn on the third day.
Jaffer struck 10 boundaries from 101 balls in 2-3/4 hours before he was caught at second slip - the last of two catches for West Indies captain Brian Lara there - off Dwayne Bravo in the final hour before the close.
The tall, languid opener shared 61 for the first wicket with fellow opener Virender Sehwag, who fell for 31, and added 63 for the second with Venkatsai Laxman to lay a solid platform for the Indians.
Laxman was not out on 28 and Indian captain Rahul Dravid was not out on 20 at the close, after the visitors had collected the last five West Indies first innings wickets for 19 runs in the space of 33 balls.
West Indies had continued to grind the Indian attack for most of the morning period with Shivnarine Chanderpaul being left stranded on 97 not out and Marlon Samuels with 87 leading their charge.
Chanderpaul struck half-dozen fours and two sixes from 213 balls in 5-1/4 hours, and Samuels smote eight fours and three sixes from 161 balls in nearly 3-1/2 hours.
Off-spin bowler Harbhajan Singh however, snared four of the last five West Indies scalps to end with five wickets - for the 18th time in Tests - for 147 runs from 44 overs to be India's most successful bowler.
Munaf Patel supported with three for 134 from 32 overs and Shantha Sreesanth and Sehwag snared one wicket apiece. India then lost Sehwag after making a brisk start to their first innings to reach 64 for one at tea.
Sehwag, who Daren Ganga dropped on 12 at gully off Pedro Collins, failed to make full use of his reprieve, and was caught at second slip for 31 edging a flat-footed drive at a delivery in Corey Collymore's first over.
It was the only success for West Indies before the break, and after the break, Jaffer and Laxman settled down to play some enterprising strokes either side of the wicket.
Jaffer reached his 50, when he square drove Collins for his eighth boundary, but driving loosely at Bravo gave West Indies their second success.
Before lunch, Chanderpaul and Samuels both completed half-centuries before the half-centuries to help West Indies continue the long, hard grind for India.
With few alarms, the pair batted through the extended morning period, after West Indies continued from their bedtime total of 420 for five.
Chanderpaul glanced Munaf Patel to deep fine leg for a single to reach his 50, and Samuels turned Anil Kumble through square leg for a single to reach his landmark.
Outside of the personal milestones for Chanderpaul and Samuels, the game meandered into a sleeper, with the West Indies committed to batting long, and India employing some negative tactics to stem the flow of runs.
Samuels and Chanderpaul carried them to 536 for five at lunch, and looked set to continue from where they left off after the interval.
Samuels lofted Harbhajan for a straight six and over wide long on for another, then Chanderpaul hit the off-spinner over wide long-off for a third in the second over after the interval that cost 19 runs.
The West Indies pair had added 156 for the sixth wicket, when Sehwag made the breakthrough.
Bowling his uncomplicated off-spin, Sehwag had Samuels caught at deep mid wicket in the third over after the lunch interval with a delivery darted into his pads, and in the following over, Denesh Ramdin was caught at square leg off Harbhajan for three pulling a shortish delivery.
Two overs later, Jerome Taylor skied a pull and was caught at mid wicket for three off Harbhajan, who removed Pedro Collins caught behind edging a doosra to slip, before the off-spinner bowled Corey Collymore through his legs to deny Chanderpaul his landmark.
The half-centuries from Chanderpaul and Samuels followed hundreds from opener Daren Ganga (128) and vice-captain Ramnaresh Sarwan (116), as well as 83 from left-handed opener Chris Gayle.
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.