India employed into attack all the four front-line bowlers, including debutant pacer Varun Aaron who surprisingly replaced very impressive Umesh Yadav, but success eluded them.
Yadav had taken a nine wickets in the second Test.
West Indies are playing without their most experienced batsman Shivnarine Chanderpaul, who could not recover in time from the calf muscle injury sustained in the second Test.
In the second over bowled by debutant Aaron, Barath hit a cover-drive to open the visitors' boundary account in front of a sparse crowd of 2,000
Brathwaite scored the second boundary when he edged Ishant Sharma.
However, the heavy outfield due to overnight dew and tight bowling by India prevented the visitors from adding to that tally till the eighth over when Barath flicked Aaron to long leg when the Jharkhand pacer drifted onto his legs.
Aaron, who also touched 140 kmph, beat the outside edge of Barath on a couple of occasions.
The lanky Sharma was shifted to the pavilion end after a six-over spell by Aaron but could not get any movement out of the track that played easy.
The first hour produced 36 runs in 12 overs with Barath marginally outpacing his partner. Barath cover-drove Sharma for his fourth four to bring up his team's 50.
Ojha, who got a bit of turn, was replaced by Ravichandran Ashwin after an economical five-over spell and Brathwaite greeted the rookie off-spinner with a boundary off a pull.
Barath hit two fours off Ashwin prior to lunch when the off-spinner, who also secured some slow turn off the track, erred outside the off.
Ojha was the most economical of the lot by giving away just 16 runs in 11 overs in two spells while Ashwin was the most expensive conceding 22 runs in six overs.
India made two changes in the morning, capping Jharkhand speedster Aaron to rest Vidarbha's Yadav.
The home team also replaced axed Yuvraj Singh with Virat Kohli, who will play his first Test since the visit to the West Indies earlier this year where he made his debut.
West Indies replaced unfit Chanderpaul with Kieran Powell who opened in the first Test at Delhi. They also brought in the fit-again Ravi Rampaul, who missed the Eden Gardens Test because of an upset stomach, for Kemar Roach.