Bravo, cousin of former West Indies batting great Brian Lara, dominated the Indian attack with a superb 166 as the tourists scored 308 runs on a batsmen's day after resuming at 267-2.
Kieran Powell (81), who replaced injured batsman Shivnarine Chanderpaul, and Marlon Samuels (61) also made merry on a batting track at the Wankhede Stadium in Mumbai.
The West Indies' batting flourished for a second successive day, with the top-six batsmen all scoring a half-century or more — only the fifth instance in the history of Test cricket.
The left-handed pair of Bravo and Powell consolidated their team's position with a 160-run stand for the fourth wicket to virtually shatter India's hopes of making a clean sweep.
The hosts lead 2-0 in the series after winning the opening Test in New Delhi by five wickets and the second match in Kolkata by an innings and 15 runs.
"I was suffering from a little bit of flu over the last couple of days and I think it took a toll on me, but nevertheless I am quite satisfied with my performance and feeling good about the team's performance," said Bravo.
"I think I am in pretty good form and hopefully it can continue. As long as you apply yourself and assess the conditions as quickly as possible, there are runs to get on this... nice batting wicket."
Bravo became debutant seamer Varun Aaron's first Test victim, caught behind while attempting to drive.scoreboard
Scoreboard
West Indies first innings: (overnight 267-2)
- A. Barath c Dhoni b Ashwin 62
- K. Brathwaite c Kohli b Ashwin 68
- K. Edwards c Dhoni b Sharma 86
- D. Bravo c Dhoni b Aaron 166
- K. Powell c Dhoni b Ojha 81
- M. Samuels c Dravid b Ashwin 61
- C. Baugh b Aaron 4
- D. Sammy c Dhoni b Aaron 3
- R. Rampaul c Kohli b Ashwin 10
- F. Edwards not out 7
- D. Bishoo not out 2
- Extras (b-8, lb-15, nb-2) 25
- Total (nine wickets; 181 overs) 575
Fall of wickets: 1-137 2-150 3-314 4-474 5-518 6-524 7-540 8-563 9-566
Bowling: Sharma 30-9-72-1 (1nb), Aaron 28-4-106-3, Ojha 48-10-126-1, Ashwin 51-6-154-4, Sehwag 16-1-61-0 (1nb), Kohli 2-0-9-0, Tendulkar 6-0-24-0.
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