Half-centuries by opener Brian Bennet (57) and Sean Williams (59) led Zimbabwe to 273 and a first-innings lead of 82.
Bangladesh reached 1-57 at stumps, with opener Mahmudul Hasan (28no) and Mominul Haque (15no), who hit 56 in the first innings, in the middle.
After bowling out Bangladesh for 191, Zimbabwe resumed on 67 without loss on Monday and looked set for a significant innings lead but Mehidy triggered a collapse.Â
Zimbabwe were initially unsettled by pacer Nahid Rana, who finished with 3-74.
Nahid struck in the day's second over. He banged one in short with searing pace and Ben Curran (18), unable to evade the extra bounce, fended it to short leg.
Bennet was also undone by Nahid's extra bounce and pace, being caught behind, with Nich Welch bowled next over by Hasan Mahmud.
Captain Craig Ervine (8) kept Williams company in a 41-run partnership, and Wessley Madhevere (24) helped him lift the score to 5-177.
Williams then found support from Nyasha Mayavo (35) but Mehidy, who was expensive in the morning session, readjusted his line and length to dismiss both as Zimbabwe lost their last five wickets for 80.
When Bangladesh came to bat in the second innings, fast bowler Blessing Muzurabani dismissed opener Shadman Islam for four but Mahmudul, dropped on 6 and 18, and Mominul survived to stumps.
