Sri Lanka amassed a mediocre 7-132 after captain Charith Asalanka had won the toss on Wednesday and elected to bat first, before Bangladesh eased their way to victory, replying with 2-133 with 21 deliveries to spare.
Bangladesh lost their first wicket first ball when Sri Lankan fast bowler Nuwan Thushara trapped Parvez Hossain Emon lbw.
But Tamim and captain Litton Das then held the Bangladesh innings together sharing a 74-run partnership for the second wicket in just 50 deliveries.
Das made 32 before being caught by Kusal Perera off spin bowler Kamindu Mendis, but Tamim joined forces with Towhid Hridoy, the pair putting on an unbroken 59-run stand that ensured a comfortable victory with 21 balls remaining.
Tamim finished with 73 not out to beat his previous best Twenty20 international score of 67 made against Zimbabwe last year. He hit six sixes and a boundary during his 47-ball innings. Hridoy remained unbeaten on 27.
It ensured Bangladesh's series win after Sri Lanka had won the first game by seven wickets and the visitors responded with a massive 83-run win to level the series
Earlier, Bangladesh off-spinner Mehedi Hasan had taken a career best 4-11 as Sri Lanka struggled.
The home side smashed 14 runs in the first over, but Bangladesh fast bowler Shoriful Islam dismissed in-form batter Kusal Mendis (6) in the last ball of the over, getting him caught at deep-square-leg by Hridoy.
Kusal Perera was out in the first ball edging Mehedi Hasan to Tamim at slip.
Dinesh Chandimal also failed, being dismissed for four, trying an aggressive shot against Mehedi only to be caught by Jaker Ali.
Asalanka (three) got stuck on the crease and played the wrong line against Mehedi to get bowled, while Mehedi caught Pathum Nissanka, the top scorer for Sri Lanka with 46, off his own bowling.
Kamindu Mendis attempted to increase Sri Lanka's scoring rate with 21 in 15 deliveries while Dasun Shanaka remained 35no off 25, including 21 of the 22 scored by Sri Lanka in their last over.
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