Nkrumah Bonner (15) and Kyle Mayers (37) dragged the Caribbeans’ second innings total to 110 at stumps on day 4 at Zahur Ahmed Chowdhury Stadium on Saturday.
With the help of Miraz, who was the first-innings hero with his maiden Test century and four-wicket haul, the Tigers now require seven wickets for a victory.
In the second innings, he failed to shine with the bat, but the spinner crushed West Indies top order by taking the wickets of openers captain Kraigg Brathwaite and John Campbell within 20 overs and first-down Shayne Moseley’s on the 25th.
Earlier, Bangladesh skipper Mominul Haque scored his 10th century in Test cricket, and his seventh ton at the Zahur Ahmed Chowdhury Stadium to become Bangladesh's highest century-maker in the longest format of cricket, surpassing Tamim Iqbal.
With it, he also achieved another milestone by becoming the joint-fastest to score 3,000 runs for Bangladesh.
The partnership was eventually broken when Liton fell for 69 with Mominul also being dismissed shortly afterwards for 115.
Mushfiqur Rahim (18) was the only other Bangladesh batsman to reach a two-digit score.
The hosts eventually declared at 223 for 8 with a lead of 394 runs to give the bowlers nearly a day and a half to carve out a victory against the West Indies.
Brief Scores:
Bangladesh 430 and 223/8d (Mominul Haque 115; Rahkeem Cornwall 3-81) lead West Indies 259 and 110/3 (Mehidy Hasan Miraz 3-52) by 285 runs.