Veteran spinner Rangana Herath continued to wreak havoc on the Galle pitch, going into the break with figures of four for 24 after he removed both Bravo, who had made a battling half century, and the experienced Marlon Samuels for 11.
Jermaine Blackwood was the other wicket to fall in the morning session. He made 11 before being caught at slip on the third attempt by Milinda Siriwardana off the bowling of the impressive Dhammika Prasad.
But barely 10 minutes before the interval he fell to an impressive catch by Dinesh Chandimal at deep mid-on.
Herath bowled Samuels after the visitors had added just four runs to their overnight score of 66-2.
The West Indies now face a mammoth task to force Sri Lanka to bat again after the hosts were all out for 484 in their first innings, thanks mainly to big centuries from Chandimal (151) and Dimuth Karunaratne (186).