New captain Brendan Taylor hit a maiden Test century for Zimbabwe and declared at 291-5 at tea on Sunday, setting Bangladesh a target of 375 to win.
Taylor finished unbeaten on 105 on the fourth day of the one-off test match at Harare Sports Club, becoming the first Zimbabwean to score a century in his first match as captain since Dave Houghton against India in 1992.
Taylor hit nine boundaries in his 175-ball innings. He shared a 113-run fifth-wicket partnership with Tatenda Taibu (59) sweeping comfortably against the spinners to restore Zimbabwe's advantage after the African side collapsed to 92-4 in the last session on Saturday.
Taibu added just six runs to his lunchtime score before he was well caught by Robiul Islam driving off the front foot from Shafiul Islam's bowling.
Taylor added 86 runs with Test newcomer Craig Ervine, who finished 35 not out.
Zimbabwe is playing in its first Test since 2005 when a fallout between players and administrators led to the country voluntarily withdrawing from the five-day game.