Zimbabwe had already clinched the series by winning the first two games but was bowled out for 151 at Harare Sports Club in 32.2 overs while chasing Bangladesh's 256-9.
Opener Anamul Haque made 76 from 71 balls for Bangladesh and No. 6 Afif Hossain was 85 not out from 81 to set up the visitors' total.
It wasn't a big score on a normally batting-friendly pitch but the hosts didn't take advantage.
Zimbabwe lost wickets in the first and second overs, and two more in the sixth over, to slip to 18-4.
At 83-9, Zimbabwe's last two batters rallied with a 68-run stand — the second-best partnership of the match — as Richard Ngarava made 34 not out and Victor Nyauchi scored 26.
The late surge was entertaining but Zimbabwe easily fell short.
Bangladesh's tour was still a success for Zimbabwe under new coach Dave Houghton, who was brought in for another spell in charge after Zimbabwe lost six straight ODI and T20 games at home against Afghanistan in June.
Zimbabwe secured its first ODI series over a fellow top-tier team since 2017 when it chased down 291 in the second ODI against Bangladesh on Sunday. It also won the T20 series 2-1 for its first-ever 20-over series win over a top-tier team.