Tajul Islam returned Bangladesh's best ever single innings haul, taking eight wickets on Monday as Zimbabwe were dismissed for just 114 in their second innings in the first Test.
Islam's 8 for 39 eclipsed the figures of 7 for 36 by fellow left-arm spinner Shakib Al Hasan against New Zealand in Chittagong in 2008, and leaves Bangladesh needing just 101 runs to win.
Zimbabwe captain Brendan Taylor did his best to hold the innings together with 45 not out but could only watch his fellow batsmen's sensational collapse around him under the sustained pressure from Islam.
Sikandar Raza was the other notable scorer with 25.
Shakib and Shahadat Hossain claimed one wicket each for Bangladesh.