After electing to bowl, Nepal - led by 4-19 from Kushal Bhurtel - gave the minnows every chance of pulling off a miraculous victory by restricting South Africa to 7-115.
Openers Bhurtel and Aasif Sheikh got Nepal's run-chase off to a solid start, combining for 35 before the former was claimed in the eighth over by Tabraiz Shamsi.
Captain Rohit Paudel's dismissal from the next ball derailed Nepal's momentum, with four of the middle order leaving the crease with fewer than 10 runs.
Wicketkeeper Sheikh proved a much more formidable foe for South Africa's bowlers, and his score of 42 - combined with 27 from Anil Kumar Sah - kept Nepal in the match.
The result came down to the final ball, with Nepal needing two runs for victory.
It was not meant to be, with Gulshan Jha (six) run out while scrambling for a bye on a delivery from Ottneil Baartman.
Shamsi led the way for South Africa's bowlers, finishing with 4-19, while Reeza Hendricks top-scored for the Proteas with 43 off 49 balls.
Joint-hosts the US qualified for the second round on their World Cup debut after their last group game, against Ireland, was washed out.
Rain meant the match was abandoned without a ball bowled, advancing the US to the Super Eight stage and automatically qualifying the Americans for the 2026 World Cup in India and Sri Lanka.
The US have used home advantage to make a stunning first impression in their first global cricket tournament.
While the Americans progressed, former champions Pakistan, along with Ireland, were eliminated.
The point from the washout was enough for the US to advance, having beaten Canada in Texas and stunned Pakistan on Long Island in the first week.
Beating cricketing superpower Pakistan in a super over was one of the greatest upsets in the tournament's history.
The 17th-ranked US join the West Indies, India, Australia, South Africa and Afghanistan in the Super Eight, with two more teams yet to qualify.
The Super Eight splits into two groups, with each team facing three games to reach the semi-finals.
New Zealand, already eliminated after losing to Afghanistan and West Indies, thrashed Uganda by nine wickets at the Brian Lara Cricket Academy in Trinidad.
The New Zealand bowlers got the ball to swerve, cut and spin throughout to dismiss Uganda for 40.
The Black Caps reached their target from a little more than five overs for the loss of one wicket, but it was little consolation for a disappointing tournament.
They will fly home whatever the result of their final group match against Papua New Guinea on Monday.
- with Reuters
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