The Indian captain gave West Indies early evidence that he'll be the wall they will have to knock over in the coming weeks if they are to overcome the visitors, as he struck 10 fours and two sixes in 105 from 102 balls that helped to earn him the Man-of-the-Match award.
Chasing 252 for victory from their allocation of 45 overs, India hit the jackpot with one ball to spare and took a 1-0 lead in the five-match series, when Mohammed Kaif drove Dwayne Bravo through extra cover for the last of his three boundaries in an undefeated 66 from 91 balls.
Dravid had featured in two important partnerships that helped set up the Indian victory. He shared 56 for the first wicket with Virender Sehwag, and consolidated his side's position with a 123-run stand with Kaif for the fourth wicket.
Chris Gayle had made hay after the sun emerged and entertained the crowd that half-filled the ground with 18 fours and two sixes in 123 his 12th ODI hundred also - from 131 balls that was the backbone of West Indies' total of 251 for six from their allocation of 45 overs.
He shared two vital partnerships that shaped the West Indies' innings after the home team was sent in to bat. He and Runako Morton put on 87 for the first wicket, then he and his captain Brian Lara added 82 for the third wicket.
Lara scored 35 from 43 balls that included four fours and one six, and Morton made 23 from 38 balls that contained two boundaries.
Dravid then went to work on wearing down the West Indies bowlers, who like many around the World, could find no way to restrict the Indian run-scoring machine.
Sehwag, whose form has been patchy of late, looked to be finding his best touch, when he was run out for 22 going for a second run, after Marlon Samuels had dropped him at long-off off the same ball in the 10th over.
India suffered another setback, when Irfan Pathan was caught at second slip off Bradshaw for one in the 12th over, and were jittery when Yuvraj Singh was caught at deep mid-off off the same bowler for 12 in the 18th over to leave them 86 for three.
Dravid was joined at the wicket by Kaif and mixing enterprising strokes with well judged running between the wickets, they put India firmly on course for victory.
Kaif benefitted from two slices of fortune that had the potential to chance the complexion of the game.
No less a person than Lara, the West Indies captain, dropped a straightforward chance at cover in the 22nd over off Fidel Edwards, who muffed a chance at deep backward square leg when Kaif had 35.
Kaif continued to plunder the bowling alongside Dravid, who drove Fidel Edwards through the hands of the mid-wicket fielder for a single to reach his landmark.
Dravid and Kaif looked like they were bringing it home, when the Indian captain was caught on the long-on boundary off Dwayne Bravo in the 39th over, but there was no stopping India.
More drama was to unfold in the closing overs. Mahendra Singh Dhoni was caught at long-on off Bravo for 18 in the 43rd over, after Jerome Taylor had delivered a maiden in the previous over, and West Indies kept up the pressure until Kaif eased the tension with the winning hit.
Earlier, the start of play had been delayed by an hour, after early morning showers drenched the Jamaica capital.
A new irrigation system installed in the outfield ahead of the 2007 ICC World Cup in the West Indies, and diligent work from the ground-staff helped the drying process and guaranteed play.
This provided Gayle and fellow opener Runako Morton with the platform to give West Indies a solid foundation and knock India's new-ball pair of Irfan Pathan and Rudra Pratap Singh out of the attack.
The two Indian Ps were either guilty of bowling too short or too full in length in their first spells, and Gayle and Morton set upon the loose bowling.
Agarkar made the breakthrough in the 14th over however, when he had Morton caught behind for 23 edging a drive playing off the back-foot.
Ramnaresh Sarwan never settled and was adjudged lbw to Harbhajan Singh for two in the 17th over to leave West Indies on 94 for two.
West Indies captain and batting superstar Brian Lara came to the wicket and consolidated his side's position in a stand of 82 for third wicket with Gayle.
Lara was never at his fluent best, but a pull through mid wicket for four perched one leg off Agarkar, and a flamboyant upper cut to third man for another boundary off Munaf Patel were a couple of memorable strokes.
Lara seemed to be getting into high gear, when he was caught at cover for 35 slapping a delivery from Patel straight into the lap of the fielder in the 32nd over.
Gayle soon arrived at his hundred, when he inside-edged a delivery from Patel past his leg-stump and into the fine leg region, but perished looking to go for broke inside the final five overs.
West Indies also lost Marlon Samuels inside the final overs in the late charge for runs, when he was run out for 10 looking for a second run and gaining no response from his batting partner Carlton Baugh Jr.
The second ODI will be contested at the same venue on Saturday, and the third will be played next Tuesday at Basseterre's Warner Park which will be hosting its first ODI.
The series concludes next Friday and Sunday at Port of Spain's Queen's Park Oval, and will be followed by a four-Test series with matches at St. John's, Gros Islet, Basseterre which is hosting its first Test, and Kingston.