Local boy Tamim entertained his home crowd with seven sixes and five fours to score 95 off 96 balls as Bangladesh eased past Zimbabwe's modest 188-6 with seven overs to play.
Tamim survived a few anxious moments at the early part of his innings and had one clear let-off when Christopher Mpofu dropped a sharp return catch with him on 24.
He never looked back after that, however, and again exploded with some powerful driving.
Tamim struck six sixes in his 154 against Zimbabwe at Bulawayo and surpassed that mark in the 29th over of the innings with a six off Dabengwa through long on.
He attempted another long bomb off the next ball but fell short as Graeme Cremer dived to take a catch at deep midwicket and deny him a deserved century.
It ended his 136-run second wicket partnership with Zunaed Siddique, who finished on an unbeaten 56.
Wicketkeeper Mushfiqur Rahim took a single off Dabengwa, taking the hosts to 189-4, which gave Bangladesh their sixth series win over Zimbabwe on home soil in the past six years.
Zimbabwe took a surprise lead in the series with a nine-run win in the first game, but Bangladesh bounced back to win the next two games by six wickets and 65 runs. The fourth one-dayer was rained off.
Zimbabwe had a disastrous start to the final game when, after choosing to bat first, they lost a wicket off the first delivery with opener Brandon Taylor suffering a freakish run out.
Some tight bowling from paceman Mashrafee Mortaza, who finished with 2-25, left them reeling at 21 for three before Tatenda Taibu and Craig Ervine staged a recovery with a 96-run stand.
Taibu made 64 for before falling victim to Shakib Al Hasan, Bangladesh's most successful bowler with 3-58.
