Hinds hit 12 boundaries in an unbeaten 73 while Gayle scored 44 in an unbroken opening stand of 118 at the close.
Fast bowlers Pedro Collins and Vasbert Drakes helped the touring team dismiss Bangladesh for 139 just after tea earlier on the day.
Collins grabbed five for 26, the second best figures of his Test career after his six for 72 against New Zealand last June.
The West Indies' paceman bowled opener Hannan Sarkar for a duck with an out-swinger and rounded off the innings by bowling Enamul Hoque for six with the first delivery of his 18th
Captain Ridley Jacobs justified his decision to put Bangladesh in to bat as the West Indies sent five batsmen to the dressing room before lunch.
Alok Kapali and Khaled Mashud added a valuable 73 in 131 minutes for the sixth wicket before skipper Mashud was bowled by a Drakes' yorker. Mashud scored 22 with two fours.
Kapali, who scored an unbeaten 89 in the third one-dayer on December 3, hit his first half-century in Test cricket. His fine 52 contained nine fours. Habibul Bashar (24) scored his 1,000th Test run.
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