Mustafizur jolted Zimbabwe with two early strikes and then returned in his final spell to polish off the tail to grab his third five-wicket haul in just nine ODIs.
His heroics left the visitors all out for 215 in 43.3 overs, in reply to Bangladesh's decent 276-9.
Bangladesh had won the first two matches by 145 runs and 58 runs to seal their fifth successive ODI series win. Imrul Kayes and Tamim Iqbal both scored 73 to set up Bangladesh's score after skipper Mashrafe Mortaza opted to bat first at the Sher-e-Bangla National Stadium.
Sean Williams scored 64 runs and Elton Chigumbura added 45 to give Zimbabwe some hope but Bangladesh did not look back after occasional spinner Sabbir Rahman broke their 80-run fourth wicket stand.
Mortaza then removed Williams, who faced 84 balls, before 20-year old Mustafizur took out Sikandar Raza and Luke Jongwe off successive balls. Tamim and Kayes earlier put on 147 runs in their opening partnership to lay the foundation for Bangladesh's total.
Zimbabwe hit back after removing the duo in quick succession but Mahmudullah smashed a rapid-fire 52 off 40 balls to help the hosts recover for a decent total. Mahmudullah, who hit five fours and a six, got a reprieve when he was on 32 as he walked assuming he was run out.
But a TV replay confirmed Zimbabwe wicketkeeper Regis Chakabva had dislodged the bails before receiving the ball, prompting the umpires to call Mahmudullah back.

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