Tamim helped Bangladesh reach 279-8 with 118 in as many balls, his seventh ODI century - the highest for his country - before the home side dismissed Afghanistan for 138 runs in 33.5 overs in a lopsided contest at the Sher-e-Bangla National Stadium on Saturday (Oct 1).
Left-arm spinner Mosharraf Hossain grabbed 3-24 runs playing his first match in more than eight years while paceman Taskin Ahmed chipped in with 2-31.
This was Bangladesh's sixth consecutive series victory in ODIs. The winning streak started in November 2014 when they blanked Zimbabwe 5-0 at home. The purple patch also included three series wins against much fancied Pakistan, India and South Africa.
During the period, Bangladesh also reached the World Cup quarter-finals in Australia-New Zealand last year. The victory in the third ODI was not assured until Bangladesh, who elected to bat first, got a commanding total for the first time in the series, which was otherwise dominated by Afghanistan.
Tamim made the most of a dropped catch early in his innings to give Bangladesh the kind of platform they were looking for, but an unexpected collapse in the middle-order denied them a bigger total.
Afghanistan skipper Asghar Stanikzai missed an easy catch when Tamim had just opened his account and the left-hander never looked back, completing his century with a single off Dawlat Zadran.
Substitute fielder Naveen-ul-Haq took the catch at long-off off Mohammad Nabi to end the onslaught of Tamim, who hit 11 fours and two sixes. Tamim got very good support from one down Sabbir Rahman, who scored his career best 65 off 79 balls and put 140 runs for the third wicket with the left-hander after the early dismissal of opener Soumya Sarkar (11).
"I should have got it in the first game. I got it in the third ODI and I'm happy," said man-of-the-match Tamim, who scored 80 in the first match.
"It wasn't the easiest wicket to bat on. I thought the ball was spinning a bit early on. The wicket was hard and dry. I took my time and finally got a hundred," he said.
Bangladesh skipper Mashrafee Moratza said the series win will give his team the desired confidence in the upcoming matches against England who arrived here on Friday.
"A lot of important series are coming up. Hopefully the boys can keep it up. I think that's a sign of improvement," he said.