West Indies made 338 for seven in their 50 overs with Bangladesh finishing on 247 for eight.
The hosts made a shaky start to be reduced to 12 for two when openers Lendl Simmons and Chris Gayle were both dismissed for six but that gave skipper Ramdin and Bravo more time at the crease.
Ramdin, who has been in fine form in the one-day game since taking over from Darren Sammy as captain, struck 169 from 121 balls, including 11 sixes and eight fours.
Ramdin's score was the joint third highest total made by a West Indies batsman in a one day international, equalling Brian Lara's 169 against Sri Lanka in 1995.
The record ODI score for a West Indian remains Vivian Richards' 189 against England at Old Trafford in 1984.
Left-hander Bravo, who should have been stumped when he was on 10, hit eight sixes of his own in his 124 from 127 balls.
Bangladesh seamer Al-Amin Hossain, who finally got rid off Ramdin, finished with 4-59.
A target run-rate of over 6.7 runs an over never looked realistic for the tourists, who had been bowled out for 70 in the second game of the series and they were quickly in trouble at two for two.
Kemar Roach pulled off a stunning one-handed catch to remove Imrul Kayes off Ravi Rampaul, who ended with figures of 4-29.
Although Bangladesh ended 91 runs short there were creditable scores of 55 for opener Tamim Iqbal and a patient 72 from skipper Mushfiqur Rahim.
(Reporting By Simon Evans in Miami; Editing by Greg Stutchbury)
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