Qatar had squandered a two-goal lead and looked to be heading out of the competition.
On a politically charged night in Doha, a Hasan Al-Haydos goal, his second of the match, secured a 3-2 win after Qatar had squandered a two-goal lead and looked to be heading out of the competition.
It means Qatar, the 2022 World Cup hosts who have endured a dreadful final round in the AFC qualifying competition, still have a remote chance of playing in Russia.
They now stand five points behind third-place Uzbekistan and will need to win both their final games in Group A -- and hope Uzbekistan lose both -- to stand a chance of qualifying.
The team that finishes third will head into play-offs for a place in Russia.
Qatar coach Jorge Fossati said after the match that his team needed "miracles" to play next year at next year's World Cup.
"We are very upset because we are convinced Qatar has the quality, unfortunately we don't have enough points," he said.
Despite the defeat, qualification for Russia is still in South Korea's own hands.
They remain a point ahead of Uzbekistan in the race for the automatic second qualifying spot in the group.
Tuesday's result could also set up a thrilling finale in the group as Uli Stielike's team play Uzbekistan in the last match.
"I take full responsibility for this result," said Stielike, who admitted his job could be on the line after a stuttering campaign. "I am the one who makes the tactical decisions."
Asked if he would consider his position following the latest setback, the German said it was not up to him.
"I have to say this is not my decision, we have to wait and see," he said.
As well as the action on the pitch, there was also plenty of drama off the pitch.
Qatar players face potential action from FIFA after they warmed-up for the match in T-shirts showing an image of the country's leader, which has become a widely-used symbol of defiance in Doha during the current diplomatic crisis in the Gulf.
FIFA does not allow teams to wear political symbols or slogans on shirts.
Qatar largely owed their victory to the industry and finishing of Al-Haydos and young forward Akram Afif.
It was a 25th-minute foul on Afif, who is the first ever Qatari to play in La Liga, which lead to Qatar's opener.
Al-Haydos took the resulting free-kick from 20 yards, scoring with a fine finish before running to the touchline to hold up a T-shirt of the Emir Sheikh Tamim bin Hamad Al-Thani to the crowd.
Afif scored Qatar's second in the 51st minute, which seemed to assure victory, but the home side's fragile defence could not stand firm.
Goals from South Korea's Ki Sung-Yueng and Hwang Hee-Chan in the 62nd and 69th minute seemed to have secured Korea a valuable point in their attempt to get to Russia.
But fine work from Brazilian-born Rodrigo Tabata set up Al-Haydos for a 74th-minute winner, prompting delirious scenes on the touchlines and in the stands.
Tottenham's Son Heung-Min went off injured in the 33rd minute after suffering a suspected broken right arm.
Stielike said after the game that Son may have suffered a fracture. "We have to have a scan, it may be broken," said Stielike. "It's not good."