Thapelo Maseko scored in the 63rd minute off a precise cross from Tshepang Moremi to earn South Africa second place in Group A behind Mexico, who won all three of their group-stage games.
"It's unbelievable, it feels like a dream," said Maseko. "We went through some rough patches, but this team showed they are capable and they're strong."
Coach Hugo Broos then produced a defensive masterclass to blunt South Korea's attack, that included former Tottenham striker Son Heung-min, and allow goalkeeper and skipper Ronwen Williams to maintain a clean sheet.
"I know how it feels but it's difficult to explain. Fantastic experience. It was so tough today but we played a very good game," South Africa coach Broos said.
"We scored the goal, and then it was 20 minutes of heart beating and waiting for the final whistle. I'm very happy for the guys."
Bafana Bafana (The Boys) will now take on co-hosts Canada, the second-place finisher in Group B, for a place in the last 16 in Sunday's match at Inglewood, California.
The African nation had failed to advance from the group stage in 1998, 2002 and as hosts in 2010.
South Africa opened their World Cup campaign with a 2-0 loss to Mexico before scoring a late goal to earn a 1-1 draw with the Czech Republic.
They entered the final group-stage game trailing Mexico, with South Korea expected to join them after starting as favourites.
South Korea opened this World Cup with a 2-1 come-from-behind win over Czech Republic and lost to Mexico 1-0 in their second game.
They could still advance in the tournament, as one of eight third-placed finishers, pending the results of other games this week.
The Taegeuk Warriors became the first Asian team to reach the knockout round in 2002, when they finished fourth overall.
South Korea also advanced out of the group stage in 2022 and 2010, both times reaching the round of 16.
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