After Shafiul Islam's strikes put Zimbabwe on back foot early in their chase of 270, Ziaur's five for 30 snuffed out any hope of a revival as the hosts were bowled out for 148.
Although Bangladesh have a healthy record at Queens Sports Club, winning six out of eight one-dayers before this encounter, the margin by which Zimbabwe lost led captain Brendan Taylor to brand the defeat a "shocker".
Despite losing Regis Chakabva and Pakistan-born debutant Sikandar Raza in opening four overs of their reply, Zimbabwe recovered through Taylor's 33 and Hamilton Masakadza's 38.
Three of those dismissals were to Ziaur, who returned later to find reverse swing and grab the wickets of Malcolm Waller and Tinashe Panyangara.
When Shingirai Masakadza's spirited knock of 33 from as many deliveries was ended by Robiul Islam, Bangladesh had a handsome victory.
Bangladesh were also indebted to Nasir Hossain, who continued his good form with the bat in rescuing Bangladesh from a mid-innings collapse.
The tourists made a steady start after being put in to bat - with Tamim Iqbal and Mohammad Ashraful putting on 65 for the first wicket - only for Shingirai Masakadza to remove the pair in his opening two overs and the Mushfiqur in his fifth.
Shingirai Masakadza's 4 for 51 was somewhat negated by the 18 wides Zimbabwe bowled, but ultimately it was the home side's batting and Ziaur's bowling that ensured a one-sided result.
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