Anamul held the hosts’ innings together amid free-fall of wickets at the other end, hitting just three fours in his 51-ball knock.
Pacer Tinashe Panyangara claimed 3-30 for Zimbabwe, who are seeking to end their tour on a high after losing the opening Twenty20 match on Friday and all three one-day internationals.
Leg-spinner Graeme Cremer and pace man Neville Madziva picked two wickets each, giving away 21 and 25 runs respectively.
Bangladesh started confidently after skipper Mashrafe Mortaza elected to bat as Tamim Iqbal and Imrul Kayes put on 34 runs in the opening stand.
Tamim miscued a pull shot off Madziva to give a catch to Sean Williams at long on for 21 and Kayes fell in the next over to Tendai Chisoro for 10.
Zimbabwe then made regular breakthroughs to keep the hosts in check.
Anamul was run out in the final over as Bangladesh settled for their lowest total in the last five Twenty20 internationals.