Gayle's 58 at the top of the order, which included five sixes and three fours, together with Bravo's comparatively serene 53 ensured there was to be no repetition of the embarrassing collapse in the first ODI two days earlier.
On that occasion the West Indies crashed to 34 for five before Kieron Pollard and Denesh Ramdin successfully resurrected the effort in pursuit of a target of 218.
The former captain bowled an immaculate line and length throughout his 10-over spell, finishing on a high when he delivered a maiden over and claimed two wickets off two balls in the 49th over to return the excellent figures of three for 39.
Al-Amin Hossain, who stole the spotlight with the ball for the visitors in the first match when he claimed four wickets, enjoyed early success again with the scalp of Kirk Edwards.
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He returned later in the innings to add Ramdin (34) to his haul although he proved more expensive in this effort, conceding 60 runs, which included sixes by Sunil Narine and Jason Holder in the last over of the innings.
Lendl Simmons struggled to get going in partnership with Ramdin, and it was only when the wicketkeeper-batsman fell that the arrival of Pollard triggered a late spike in the run-rate.
Pollard smashed a six and two fours in 26 off 20 balls before playing on to Mashrafe.
Simmons got to 40 but occupied 61 deliveries in doing so, his painstaking progress keeping the final total at a level which Bangladesh will feel may be within their reach depending on how well they start their reply.