Gayle top-scored with a typically belligerent 72, Bravo hit an attractive 63, and Ramnaresh Sarwan laboured over 57 to help West Indies reach 318 for six, in reply to India's first innings total of 241, when stumps were drawn on the second day.
Virender Sehwag dismissed Bravo and left-hander Shivnarine Chanderpaul in the last 1-1/2 hours to be the pick of the Indian bowlers with two wickets for 32 runs from 12 overs.
Munaf Patel supported with two for 60 from 21 overs and leg-spin bowler Anil Kumble took two for 85 from 26 overs.
West Indies could have been in a much stronger position had they not lost three wickets after tea, two to Sehwag, following half-centuries from Gayle, Bravo, and Sarwan.
Sehwag however, intervened with his uncomplicated off-spin to stem the flow of runs, and stop a fifth-wicket stand between Bravo and Chanderpaul on 73.
With his frontline bowlers unable to make headway, Indian captain Rahul Dravid turned to Sehwag and he removed Chanderpaul caught behind for 24 when the batsman essayed a cut and got an under edge.
He then snared Bravo four overs later, when the young all-rounder was deceived by the flight and bounce of a delivery and was stumped after he struck 11 boundaries in 92 balls from 1-3/4 hours.
Earlier, the runs had flowed for West Indies, with Gayle and Sarwan marauding India's three fast bowlers, after the home team suffered an early setback.
Daren Ganga, who did not feature in the preceding five-match One-day international series between the two sides, was adjudged lbw for nine off Patel in the fourth over of the innings.
Gayle with typical savagery then jumped on the new-ball pair of Sreesanth and Patel, as Sarwan played smoothly at the other end. Gayle's first scoring stroke was a top-edged cut over the slips to third man and he repeated the stroke in the next over from Sreesanth.
He lofted the first ball of Sreesanth's fifth over for a straight six and collected two boundaries square of the wicket off the last two deliveries in an over that cost 17.
Sarwan cut the debutant fast bowler's first delivery past gully for four, and after being struck by a full toss on the thigh pad, drove him exquisitely through mid-off for another four to help West Indies reach 81 for one at lunch.
After the interval, West Indies continued their assault on the Indian fast bowlers in particular, but were slowed down by the dismissals of Gayle and fellow left-hander Brian Lara in action-packed afternoon period.
Gayle was caught at slip off Kumble for 72 edging a forward defensive probe, after he added 119 for the second wicket with Sarwan, and Lara was caught at point off Patel slicing a drive for 18 to leave West Indies on 159 for three.
Gayle, in particular, was typically carefree. He arrived at his 50, when he collected successive boundaries off Patel in the third over after the interval. The first he drove powerfully past mid-off; the second, he steered to third man.
Gayle also enjoyed consecutive boundaries in Kumble's first over, but the wily leg-spinner eventually pinned him down, and made the breakthrough for India, when he removed the West Indies opener, who struck 13 fours and one six from 91 balls in 2-1/4 hours.
A few rasping strokes from Lara suggested he was finding his range and would treat the crowd to another gem of an innings, but he drove loosely at a delivery from Patel outside the off-stump and was dismissed to leave the crowd disappointed.
Through it all, Sarwan kept his composure and eventually reached his 50 when he turned Vikram Singh through backward square leg for a single to continue his purple patch of scoring against the visitors, and reach 4,000 runs in his 60th Test.
West Indies reached 144 for four at tea, but Sarwan was adjudged lbw to a flipper from Kumble - the first ball after the break - after striking nine fours from 110 balls in about three hours of batting.
Bravo came to the wicket and with Chanderpaul consolidated West Indies' position with their partnership.
Bravo was prepared to be more adventurous than the sedate Chanderpaul and some elegant strokes on the leg-side, where he is strongest, were a joy to behold.
He gave West Indies the lead when he straight drove Patel for the eighth of his 11 boundaries, but he survived a close shave two close shaves, when he turned balls from Kumble through the legs of short-leg fielder Mohammed Kaif.
Sehwag broke through the partnership between Bravo and Chanderpaul, much to the delight of the visitors, leaving Denesh Ramdin, on 22 not out, and Ian Bradshaw, on 15 not out, to carry West Indies the rest of the way.
On resumption, tailenders Shantha Sreesanth and Patel frustrated West Indies for about half-hour. They added only six runs though, before Fidel Edwards bowled Patel for a 28-ball duck.