Namibia upset the defending champions to join host Bangladesh in the quarterfinals from Group A.
South Africa, batting first, was reduced to 60 for eight by the half-way stage following a four-wicket haul by Michael van Lingen. But Willem Ludick hit 42 in a ninth-wicket stand of 55 runs with Lutho Sipamla as the young Proteas added a further 76 runs.
Defending a modest 136 for nine in 50 overs, South Africa claimed a wicket off the first ball and continued to dismiss batsmen regularly, but Lohan Louwrens stood in the way as he steered Namibia home with an unbeaten 58.
Namibia's captain Zane Green said he was delighted with the win and said he was not surprised that his team had made the knockout rounds.
"We were positive from the start of the tournament because we knew we had the team to go far," said Green. "Our aim was to take it one game at a time and that is what we will continue to do.
"We want to take the positive energy from this game into our next match against Bangladesh. And then take it from there."
"Our top-order failed badly," he said. "If the boys are honest with themselves, they will realise they played some loose shots. We should have got more than 136 for sure.
"We have let our country down by not qualifying for the quarter-finals, but we will definitely try to finish ninth .... That's the best we can do."
In other matches, Bangladesh beat Scotland by 114 runs in Cox's Bazar for its second win and now clashes with Namibia in the last league match on Tuesday to decide Group A winner.
England brushed aside Zimbabwe by 129 runs in Chittagong to top Group C with a third successive win. England will now wait the second-placed team from Group B in the quarter-final on 7 February in Mirpur.
The West Indies romped to a 262-run win over Fiji in another Group C match in Chittagong. It now faces Zimbabwe on Tuesday to determine the second qualifier from the group with the winner meeting Group B champion (Pakistan or Sri Lanka) in Fatullah on February 8.
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