Rahul Dravid compiled an assured 112 to help India set the West Indies a target of 326 to win in the first test at Sabina Park.
India was bowled out on the stroke of tea for 252 and the West Indies closed day three at 131-3, still needing 195 for victory.
Adrian Barath (38) and Lendl Simmons (27) launched the chase in positive fashion by putting on 62 for the first wicket in 11 overs before India struck back strongly.
Ishant Sharma (2-41) and Praveen Kumar (1-27) started the wobble before an unbroken fourth-wicket stand of 51 between Darren Bravo (30 not out) and Shivnarine Chanderpaul (24 not out) boosted the West Indies' chances.
Earlier, the 38-year-old Dravid anchored the visitors, his innings spanning 403 minutes and supplying a crucial lead.
The former captain was delighted that his century set up the chance of a victory.
"It was a satisfying one. Obviously, satisfying in the context of the game," Dravid said of his fourth century against the West Indies. "We needed to get a decent score to put them under some pressure."
Dravid, dropped on six the previous evening at second slip by West Indies captain Darren Sammy, cracked 10 fours off 274 deliveries before he was last man out.
Sammy ended with the best figures in the West Indies attack, claiming 4-52. Leg-spinner Devendra Bishoo took 4-65.
Dravid got good support from Amit Mishra, whose adventurous 28 was the next best score in India's innings.
The pair added 56 for the ninth wicket to lift the visitors from the uncertainty of 183-8 just after lunch.
"Amit, I thought, showed a lot of character, a lot of guts today and really stuck it out there with me," Dravid said, "That was great."
And Dravid remained confident India could secure victory on the fourth day.
"I think it's a good score. Obviously this partnership is going to be the one that we need to break early in the morning. Chanderpaul is obviously a good player, Darren Bravo can play," he said. "There is still something in the wicket."
The West Indies struck early on day three when Dravid lost his overnight partner Virat Kohli for 15 in the fourth over to make the total 100-4.
The righthander, on debut, gloved a Fidel Edwards lifter to wicketkeeper Carlton Baugh down the leg side.
Dravid soon passed his 50, with his sixth boundary, and forged a fifth-wicket stand of 48 with Suresh Raina.
Lefthander Raina compiled 27 off 53 balls with three boundaries before he swept at Bishoo and deflected a catch to leg slip off Baugh's thigh.
Captain Mahendra Singh Dhoni oozed positive intent from the time he arrived at 148-5, lashing a six and a four off Bishoo.
But after Dhoni reached a run-a-ball 16, Bishoo got his revenge when the right-hander slapped a short ball straight to cover as India slipped to 166-6.
India was further undermined on resumption by Sammy, who claimed the wickets of Harbhajan Singh (5) and Praveen Kumar (0) in the space of three balls.
Harbhajan was adjudged lbw to an inswinger, while Kumar played around a full length delivery and was bowled at 183-8, ahead by just 256.
But Dravid remained resolute and found an equally determined partner in Mishra.
Mishra played some authentic strokes, lashing four fours off 60 balls in 77 minutes before he upper cut Sammy straight to third man.
After Dravid's dismissal, Barath and Simmons went on the attack to open the West Indies chase and shared a boundary-studded opening stand of 62.
Kumar made the crucial breakthrough when Barath, who had earlier edged through a vacant third slip, again drove and this time found the man just placed there. Barath lashed four fours and two hooked sixes off Sharma off 36 balls.
Sharma soon dented the hosts further with the dismissals of Simmons and Ramnaresh Sarwan within three balls.
Simmons, after hitting three fours and a six off 41 balls, was bowled by a fine leg-cutter, while Sarwan drove loosely and was brilliantly caught high at gully by Virat Kohli.
But Bravo and Chanderpaul kept West Indies alive in the final hour and a-quarter with their half-century stand.
