Bangladesh, chasing a huge 508-run target, lost opener Imrul Kayes and Shahriar Nafees before reaching 73-2 in their second innings at tea on the penultimate day.
Pacer Fidel Edwards, who rocked Bangladesh with five quick wickets in the first innings, struck in his second over when he had Kayes (nine) caught by Kirk Edwards at first slip.
Seamer Darren Sammy then held a return catch to dismiss Nafees (18) in the last over before the break.
Tamim Iqbal, dropped on 12 by Marlon Samuels at point off Fidel Edwards, was unbeaten on 42.
Left-handed Bravo, aged 22, smashed five sixes and 12 fours in his 297-ball 195 before falling in the afternoon, top-edging a delivery from debutant left-arm spinner Suhrawadi Shuvo to wicket-keeper Mushfiqur Rahim.
He had hit a six off the previous ball before going for another big shot in a bid to complete his double-century. He added 143 for the fifth wicket with Shivnarine Chanderpaul, who remained unbeaten on 59 with six fours.
The West Indies added 176 runs to their overnight total of 207-3 before declaring their second innings closed at 383-5 after Bravo's dismissal.
Bangladesh were bowled out for 231 in their first innings yesterday in reply to the West Indies' 355.
Bravo, who completed his maiden Test hundred off the last ball today, reached the 150-mark in the last over of the morning session with a four off Shuvo.
He dominated the Bangladeshi attack for more than a session, adding 95 to his overnight score of 100. He also added 56 for the fourth wicket with nightwatchman Kemar Roach (12). Roach fell in the day's 11th over when he was caught by Naeem Islam at long-off off Shuvo, who finished with 3-73. Left-arm spinner Shakib Al Hasan took 1-79 off 21 overs.
The opening Test ended in a draw in Chittagong.