Yoon grabbed the winner at the end of the first period of the extra session in the fifth straight quarter-final meeting between the sides at the continental showcase event.
Australia left it even later in the day's other game, with Harry Kewell heading the 1-0 winner against title holders Iraq in the 117th minute.
The Socceroos now have a semi-final against Uzbekistan, who beat Jordan 2-1 Friday, while South Korea will face Japan, who got a last-gasp 3-2 over hosts Qatar Friday.
South Korea, who have twice won the showpiece of Asian football, dominated a scrappy first half against Iran without creating any clear-cut chances.
The World Cup finalists should have taken the lead midway through the second half when Ji Dong Won headed straight at goalkeeper Mahdi Rahmati.
South Koreans had a lucky escape when Mohammad Reza Khalatbari, Karim Ansari Fard and Pejman Nouri were all involved in a goalmouth melee, that saw five players entangled on the penalty spot, but the ball landed safely in Jung Sungryong's arms.
South Korea dominated extra time with their short-passing play and got the winner in first-half injury time when substitute Yoon, who came on in the 81st minute, found himself in space and unleashed a powerful shot that gave Rahmati no chance.
Iran then threw caution to the wind, but could find no way through the tight Korean defence.
Iran coach Afshin Ghotbi said that his players had not played as well as they are capable of but also admitted "that we lost to a very good Korean side".
In the earlier game Kewell's late goal saw Australia move to within two victories of their first major football title. They went out in the quarters in their Asian Cup debut four years ago.
"I am particularly happy for Kewell that he got the deciding goal," said Australia's German coach Holger Osieck.
"We had more parts of the game in 90 minutes, we had good structure and created many chances. We should have done our job during that time. (But) Iraq came back very strongly in extra time."
The defending champions had the Australian backline in trouble in the 16th minute when Nashat Akram and Younus Mohammed combined, but Luke Wilkshire manage to clear for a corner.
Matt McKay should have given Australia the lead shortly before the break after being sent through by Brett Holman, but with just goalkeeper Mohammed Kassid to beat, shot straight into the arms of the keeper.
Iraq were the more enterprising side at the start of the second half and in extra time as well, although they were somewhat lucky in the 96th minute when Mile Jedinak headed over a goal-bound overhead kick from Sasa Ognenovski.
The only goal of the game finally came three minutes from the end of an extra-time that Iraq completely dominated Matt McKay floated a long ball into the area onto the head of Kewell and the former Liverpool player made no mistake to beat goalkeeper Mohammed Kassid.
"We did everything in the game, but are left with nothing. In the end it was one cross and one header," said Iraq coach Wolfgang Sidka.