Son Heung-min was moments away from an early return to Tottenham Hotspur
Australia had looked set for victory thanks to Craig Goodwin's volley from Nathaniel Atkinson's cross.
But Son was fouled by Lewis Miller and Hwang Hee-chan's 96th-minute penalty sent the game to extra time.
Tottenham's Son was the difference in the added 30 minutes, as he curled a free-kick past Mat Ryan in Qatar.
Australia ended with 10 men after Aiden O'Neill was sent off by the video assistant referee for a dangerous foul.
South Korea, who last won the Asian Cup in 1960, were on the brink of elimination for the second round in a row.
In the last 16, Cho Gue-sung's 99th-minute equaliser denied Saudi Arabia with the Koreans winning on penalties.
This win could help relieve some pressure on Jurgen Klinsmann, who was appointed a year ago and is unpopular with the Korean fans.
Graham Arnold's Australia squandered several good chances to double their lead, and potentially wrap up the game, with Martin Boyle being denied by a fine Cho Hyun-woo double save and Mitchell Duke missing two good chances.
Jordan will be South Korea's opponents in Tuesday's semi-final after a 1-0 win over Asian Cup debutants Tajikistan.
Earlier on Friday, Jordan made history as they reached the semi-finals for the first time. Vahdat Hanonov's own goal, as Abdallah Nasib's header went in off his chest, saw off surprise package Tajikistan.
Iran face Japan and hosts Qatar play Uzbekistan in Saturday's other two quarter-finals.
-- Courtesy of BBC Sport