Fast bowler Taskin took five for 35 as South Africa were bowled out for 154, their lowest total against Bangladesh.
Visiting captain Tamim then scored 87 not out off 82 balls to take Bangladesh to victory with 23.3 overs to spare, sharing a century opening partnership with Liton Das (48).
Bangladesh won the World Cup Super League series by two matches to one, consolidating their place at the top of the table.
South Africa were left languishing in ninth place and in danger of missing one of the eight automatic qualifying places for the 2023 Cricket World Cup.
Bangladesh had lost all 19 matches across all formats against South Africa on their home soil before winning the series opener by 38 runs at the same venue on Friday.
South Africa won the second match by seven wickets in Johannesburg but Bangladesh dominated Wednesday’s series decider.
South Africa made a fast start, with Janneman Malan and Quinton de Kock putting on 46 for the first wicket inside seven overs.
But Taskin bowled superbly after off-spinner Mehidy Hasan made the first breakthrough by dismissing De Kock.
Taskin, 26, produced his best performance since taking five for 28 on debut against India on debut in Mirpur in 2014. He bowled at a lively pace in three spells and troubled all the batsmen.
With South Africa defending a low total, Keshav Maharaj dropped Litton Das at backward point off Kagiso Rabada in the first over of the Bangladesh innings before Das had scored.
Das and Tamim took full advantage of the let-off, putting on 127 for the first wicket.
Brief scores:
South Africa 154 in 37 overs (J. Malan 39; Taskin Ahmed 5-35, Shakib Al Hasan 2-24) v Bangladesh 156-1 in 26.3 overs (Tamim Iqbal 87 not out, Liton Das 48).
Result: Bangladesh won by nine wickets.
Series: Bangladesh won the three-match series 2-1.
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