Dravid, showing all the patience and sublime skill built up over a 15-year Test career, batted for almost six and a half hours to compile 112 in a low-scoring match where both teams have struggled to score freely.
His performance enabled the tourists to post a second-innings total of 252 and set West Indies a daunting target of 326 to win the series opener.
West Indies racked up 131 to reduce their target to 195 off by the time stumps were drawn on the third day but lost three wickets in the process, leaving India in control of a match seemingly destined to finish early.
Dravid's hundred was his 32nd in Tests and his fourth against West Indies and the 38-year-old provided his younger teammates with a lesson in resilience as he cautiously navigated his way to triple figures on a pitch offering assistance to the spinners.
None of the other batsman got to 30 while Darren Sammy and Devendra Bishoo were the pick of the West Indian bowlers, capturing four wickets each.
India started the third day on 91 for three but lost Virat Kohli (15), Suresh Raina (27) and Mahendra Singh Dhoni (16) in the morning session as Dravid took his overnight score from 45 to 69.
Content to bat cautiously and occupy the crease, Dravid accelerated in the middle session, reaching his ton off 252 balls before becoming the last wicket to fall.
West Indies made a flying start to their run chase when Adrian Barath and Lendl Simmons brought up their 50 partnership in just 55 balls before India made three quick strikes, removing Barath (38), Simmons (27) and Ramnaresh Sarwan, for no score, to leave the home side reeling at 80 for three.
Darren Bravo (30 not out) and Shivnarine Chanderpaul (24 not out) added 51 in an unbroken stand fopr the fourth wicket to guide the home team to the close without further casualties and give them some hope of reaching their target on Thursday.