The hosts decided against enforcing the follow on in St Lucia, building a 427-run lead at stumps on Monday.
They bowled Bangladesh out for 161 to take a 219-run first-innings lead before reaching 208-4 at the close.
Veteran Shivnarine Chanderpaul (63 not out) continued his good form and the left-hander will be joined by Jermaine Blackwood (43) as they look to bat on early on day four.
Bangladesh resumed at 104-7 and Mahmudullah added some respectability to the total as he reached 53.
Sulieman Benn (2-19) and Shannon Gabriel (1-49) took care of the tail.
After opting to bat on in their bid for a 2-0 series sweep, the Windies made a good start to their second innings.
Kraigg Brathwaite (45) and debutant Leon Johnson (41) put on 76 for the first wicket.
Kirk Edwards (two) and Darren Bravo (seven) fell cheaply, but Chanderpaul and Blackwood put on an unbeaten 108-run stand.
Chanderpaul, scoring uncharacteristically quickly with 63 off 98, has made three consecutive half-centuries and is aiming for his first ton of the series.