The left-arm spinner ripped through Zimbabwe on his way to his 12th five-for as the tourists were dismissed for 240.
At stumps, Bangladesh reached 27-1 to trail by 213. Mominul Haque was on 14 and Shamsur Rahman on 8.
Tamim Iqbal (7) was the only batsman out after he was undone by a rising delivery off Tinashe Panyangara, fending the ball to Masakadza who stretched high to hold the catch.
Shakib was returning to the national team after serving a three-month ban for his "severe attitude problem". He behaved impeccably and his haul meant he has now taken five wickets against every test nation.
Most Zimbabwe batsmen started well but failed to carry on, with Shakib hastening their downfall. He was helped by newcomer Jubair Hossain who returned figures of 2-58 .
Opener Sikandar Raza was top scorer with 51, including six fours, while Crag Ervine made 34.
Elton Chigumbura (29) and captain Brendan Taylor (28) also contributed.
Raza and Taylor added 52 for the third wicket to help Zimbabwe settle before Bangladesh got back into the match with three quick wickets after lunch.
After Raza, Craig Ervine held Zimbabwe together but he too fell with a needless shot to hasten his side's collapse. Shahadat Hossain and Taijul Islam claimed one wicket apiece for Bangladesh.
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