The Socceroos looked headed for the semi-finals after Craig Goodwin gave them the lead at the end of the first half and South Korea struggled to find a response.
But Hwang Hee-chan equalized from the penalty spot in the 96th minute before Tottenham star Son Heung-min curled home a free-kick for the winner in the first period of extra time.
Australia were left kicking themselves -- they should have doubled their lead 10 minutes after the restart when Mitchell Duke blazed over.
Duke spurned another opportunity when he headed wide in the 84th minute, and the Koreans made them pay.
Australia finished the match with 10 men after Aiden O'Neill was red-carded for lunging at Hwang.
"We were up 1-0, we had chances for 2-0, 3-0, and if you don't take your chances you get punished," said Socceroos coach Arnold.
"It's up to the individual to take the chances."
Arnold said he had "learned a lot from the Asian Cup" but admitted the defeat was hard to take.
"We're devastated at the moment, quite emotional for all the players and the staff," he said.
"For the first 90-odd minutes we did very well until giving that penalty away.
"It's been a great tournament for a lot of my players."
Lewis Miller gave away the penalty at the end of second-half injury time when he took down a surging Son.
Miller was again at fault when he fouled Hwang for the free-kick that Son dispatched for the winner.
"I put my arm around him and give him a hug and tell him that these things are lessons in life," said Arnold.
"You learn from these type of things and you move forward."