The Dutch, who battled through the preliminary round to reach the Super 12, held the Asian nation to 144-8 with Paul van Meekeren becoming his country's joint all-time T20 wicket-taker along the way.
But they got off to a horror start in the run chase, losing two wickets in the opening two balls to Ahmed, then slumped to 15-4 in cold conditions at Hobart to put them on the back foot.
Colin Ackermann with 62 off 48 balls played a lone hand, making his maiden T20 half-century as partners fell around him and the Netherlands were all out for 135 on the final ball.
Veteran Bangladesh captain Shakib Al Hasan has played in every edition of the World Cup since its inception in 2007 and this was his first Super 12 victory.
"It was an important win, I've played all editions but haven't won and that was on the back of my mind," he said.
"Taskin has been a good bowler for us, he has the experience and pace. Most of the fielders in our team are agile and quick, in the field we can save five to 10 runs, and that could be the big difference."
The defeat was a heavy blow with two points desperately needed in a group that features heavyweight opponents India, Pakistan and South Africa.
But it was a relief for Bangladesh, who qualified directly for the Super 12 but have suffered from poor form since the 2021 World Cup, playing 19 T20 internationals, losing 14, before meeting the Dutch.
Needless run-outs
"Our bowlers were quite amazing, but our run-outs (Max O'Dowd and Tom Cooper) were poor," said Netherlands skipper Scott Edwards.
"Hopefully we can bring in a better performance with the bat, looking forward to the next challenge (against India)."
After being sent into bat, Bangladesh plundered 12 off Fred Klaassen's opening over and kept the pressure on as the bowlers struggled to maintain their line.
With temperatures hovering around 12 Celsius (54 Fahrenheit), the Dutch took time to warm up but when Van Meekeren switched ends he found extra pace and Soumya Sarkar (14) mistimed a pull shot to Bas De Leede.
Spinner Tim Pringle was brought on and ended fellow opener Najmul Shanto's innings on 25 in the next over as Bangladesh faltered.
Litton Das was out for nine, slapping a Van Beek ball to Tim Cooper at mid-off, before Al Hasan fell four balls later, caught in the deep off Shariz Ahmad to leave Bangladesh at 70-4.
Van Meekeren struck again, bowling Yasir Ali with a yorker to draw level on 58 wickets with Pieter Seelaar as the Netherland's top T20 wicket-taker.
Afif Hossain (38) and Nurul Hasan (13) put on 44 before De Leede removed them both in the same over.
The Netherlands had the worst possible start, as Vikramjit Singh edged Ahmed to Ali at slip then De Leede got a nick to wicketkeeper Hasan before Ackermann successfully fended off the hat-trick ball.
Two needless run outs within three balls followed with O'Dowd (8) and Cooper (0).
They reached 59-4 before captain Edwards was undone by the spin of Al Hasan then Pringle (1) departed to Hasan Mahmud.
After a brief rain delay Ackermann, who hit six fours and two sixes in his 48-ball knock, gamely kept up the chase before he was the ninth man out trying to clear the boundary again.
View the scorecard from Bangladesh's win over the Netherlands on Flashscore.
