The Dutch, who posted the lowest T20 international score of 39 in Monday's rout by Sri Lanka, looked ready to pull off the biggest upset of the tournament before imploding when set for victory.
Chasing 146 to win in Chittagong, opener Steven Myburgh (51) gave them a flying start as he thumped eight fours and two sixes in a 28-ball knock.
However, after Myburgh played a ball from spinner JP Duminy on to his stumps the Dutch produced a series of rash shots and lost wickets at regular intervals.
Leg-spinner Imran Tahir claimed four for 21 while pace spearhead Dale Steyn picked up two for 19.
"We obviously let this one go," said Dutch captain Peter Borren at the presentation ceremony. "We played across the line to Tahir and you've got to play him a bit straighter.
"It's a bit of a shame our batters couldn't get the job done. As miffed as I am with the fact that we didn't get over the line I am also proud with the way we stepped up after our game the other night."
Earlier, Ahsan Malik's five for 19 restricted South Africa to 145 for nine after they were put in to bat.
Opener Hashim Amla gave South Africa a brisk start with 43 off 22 balls containing seven fours and a six.
They then lost wickets in clusters against Malik's medium pace bowling and struggled for momentum towards the end of the innings.
"We are making it hard for ourselves. It's a great sign when your side isn't playing at 100 percent and you still manage to get across the line," said South Africa captain Faf du Plessis, also referring to the tense two-run win over New Zealand.
"We are playing at about 50-60 percent...that was a great fightback. Netherlands played fantastically today.
"You have to give them credit. Obviously they didn't pull through a game they should have won but they stepped up to the plate."
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