Having won the first two games, the hosts lost the toss and, after being put in, were grateful for Mahmudullah's innings of 52 from number seven.
Spinner Sunil Narine took four for 37 to give his side a good chance of the win they need if they are to turn around the five-match series.
After a stand of 57 between openers Tamim Iqbal (22) and second-Test centurion Anamul Haque (33), Narine did the early damage by removing Tamim and Naeem Islam (four) in his first over and Anamul in his next.
Nasir Hossain (six) became Veerasammy Permaul's first ODI victim before the Tigers showed some overdue resistance with a stand of 32 between Mushfiqur Rahim and Mominul Haque.
Chris Gayle removed the latter for 12, driving loosely to Kieron Pollard at short cover, but Mushfiqur was joined by Mahmudullah in the second half-century stand of the innings.
Wicketkeeper Mushfiqur advanced to 38 before top-edging a sweep at Permaul (two for 40) straight to rival captain Darren Sammy but Mahmudullah hit three fours and a six in his half-century.
He put on 39 with Sohag Gazi before driving Dwayne Smith to Sammy at mid-off. Gazi reached 30 and was the last man out when he crashed the first ball of the final over, bowled by Sammy (two for 46), straight to Gayle at extra cover.
- Sports & Recreation
- Sunil Narine
- Tamim Iqbal