Shakib, who was playing his first T20 match in almost a year, wrapped up Zimbabwe's innings by dismissing Blessing Muzarabani and Richard Ngarawa in the final over.
Zimbabwe, who elected to bat first after winning the toss, gave a great start to Bangladesh when openers Tanzeed Hasan Tamim and Soumya made a partnership of 101 runs for the first wicket.
Luke Jongwe broke the opening partnership by dismissing Tanzid for 52 runs in 37 balls and then Soumya's wicket fell for 41 runs in the same over.