Captain Raza hit an unbeaten 72 off 46 to guide Zimbabwe to 158-2 after they restricted Bangladesh, who won the first four matches of the series, to 157-6.
Bennett struck 70 off 49 balls after picking up 2-20 with the ball at the Sher-e-Bangla National Stadium.
Zimbabwe lost opener Tadiwanashe Marumani for one, but Bennett and Raza added 75 runs for the second wicket to put the visitors firmly in control.
Mohammad Saifuddin dismissed Bennett, who struck five fours and as many sixes, but Raza defied a muscle injury to see Zimbabwe to victory.
He struck six fours and four sixes in his unbeaten knock before Johnathan Campbell hit the winning single with nine balls to spare.
Mahmudullah Riyad earlier hit 54 off 44 balls for Bangladesh.
The right-hander, who struck six fours and a six, put on 69 runs with skipper Najmul Hossain Shanto to help Bangladesh recover from 15-3 after they were asked to bat.
All-rounder Shakib Al Hasan added 21 off 17 balls after Najmul's departure for 36 runs after facing 28 balls.
The hosts were in trouble when Blessing Muzarabani removed opener Tanzid Hasan Tamim for two and Brian Bennett struck twice in successive overs.
Bennett had opener Soumya Sarkar caught by Sean Williams at backward point for seven before he forced Towhid Hridoy to edge behind for one.
Mahmudullah launched a counter-attack, cracking three fours off the first four balls he faced.
Wellington Masakadza dismissed Najmul to break the partnership, but Shakib's brief cameo kept the scoreboard ticking.
Luke Jongwe brought a premature end to Shakib's innings, but Jaker Ali Anik hit an unbeaten 24 off 11 balls.
Muzarabani took 2-22 for Zimbabwe.
