Jaffer was undefeated on a Test-best 113 to be the cornerstone on which India have built a second innings total of 215 for two to earn a lead of 85 when stumps were drawn on the third day.
Jaffer, whose batting on India's previous trip to the Caribbean helped to earn him a place on this tour, has struck one-dozen boundaries from 222 balls in close to three hours of batting.
He shared three successive half-century partnerships that has established a solid base for the Indian innings.
He and fellow opener Virender Sehwag shared 72 for the first wicket, he then added 75 for the second wicket with Venkatsai Laxman, and Indian captain Rahul Dravid joined him to navigate through two hours before the close in which they put on 68, unbroken, for the third wicket.
Sehwag, who scored 41, and Laxman, who made 31, were the only victims for India, after the visitors dismissed West Indies for 371 to concede a first innings deficit of 130.
Dravid, also known as "the Great Wall of India," batted with typical aplomb to be not out on 21 at the close.
Comfortable against pace and spin, and assertive with strokes either off the front or back foot, Jaffer formally put India in the black, when he cut left-arm chinaman bowler Dave Mohammed - in his sixth over - to short third man for a single.
He had reached 50, when he drove Dwayne Bravo through mid-off for his sixth boundary in the last half-hour before tea, and arrived at his milestone, when he drove Ian Bradshaw through backward point for his 11th boundary.
Along with the thrill of dismissing West Indies, India had enjoyed the sight of a couple enterprising strokes from Sehwag before lunch.
The shaven-skull Indian opener cut a short, wide ball from Fidel Edwards over backward point for six, disturbing a cluster of young children in the bottom deck of the Andy Roberts Stand, and whipped Corey Collymore through mid-wicket for another boundary in the brief period before the interval.
After lunch, there was an intriguing duel between batsman and bowler, and Sehwag was fortunate that close fielders on the leg-side twice failed to get their hands to the ball, when he fended short, rising deliveries from Edwards.
Edwards sought to exploit Sehwag's apparent weakness to the short ball into the rib cage, and on 15, the batsmen turned a rib-tickler over the head of forward short leg fielder Ramnaresh Sarwan, whose diving effort failed by inches.
On 22, another ball into the ribs was fended away one foot short of leg-slip fielder Shivnarine Chanderpaul, much to the chagrin of his West Indies teammates and the crowd that half filled the ground nestled in the heart of the Antigua capital.
A straight drive for four in the next over - bowled by Ian Bradshaw - would have eased Sehwag's nerves, but he would have been entirely relieved when Edwards limped off the field with a stiff hamstring coming to the end of his sixth over.
Things got relatively easier for the Indian batsmen, after the West Indies' fastest bowler left the field. He remained off for the rest of the day, and would seek medical treatment to determine the severity of his injury.
For a time though, Bradshaw and Collymore plugged away, and the pressure built and eventually took its toll on Sehwag. Trying to free the shackles, he was caught at first slip off Collymore.
Laxman came to the wicket and with Jaffer carried India to tea on 121 for one, but he was just beginning to get into his groove after the break, and India were only 17 runs to the good, when Mohammed made the breakthrough.
Laxman was caught at mid-on for 31, when he top-edged a sweep at a delivery from Mohammed that turned and bounced sharply.
Earlier, the home team continued from their bedtime total of 318 for six, and India claimed the second new ball right from the start.
India's three fast bowlers - Munaf Patel, Shantha Sreesanth, and Vikram Singh - all bent their backs to extract plenty of bounce and movement from the pitch, and West Indies were dismissed about half-hour before lunch.
Patel was the pick of the Indian bowlers with three wickets for 80 runs from 28 overs, and Anil Kumble collected three for 86 from 27.3 overs.
Sehwag with two for 32 from 12 overs, and Vikram Singh with two for 61 from 15 overs gave support. Denesh Ramdin was caught behind for 26 when he edged a forward defensive probe off Patel in the fifth over of the morning to end his crucial seventh-wicket stand with left-hander Bradshaw on 49.
West Indies were still 10 runs short of the psychological advantage of a 100-run lead, when Mohammed entered to play with some adventure to formalise the landmark when he slashed Patel over the slips for a boundary.
Bradshaw, at the other end, was showing plenty of application, but he became absorbed by the extravagance of his teammate, and Vikram Singh got him when he was caught at backward point for 33 fending away a short, rising delivery.
After his departure, it took India another half-hour before ending the West Indies' innings, as the last three wickets fell for 12 in the space of 28 balls.
Edwards was caught behind for four fending away Vikram Singh, and Collymore was adjudged lbw to Kumble to leave Mohammed not out on 19.
The four-Test series between the two sides also features matches at Gros Islet (June 10-14), Basseterre (June 22-26), and Kingston (June 30-July 4).