Hiddink said he felt that his side had gained control of the
match when they gave away possession in the 75th minute, allowing Oliver Neuville to feed Michael Ballack for the winner.
"We killed ourselves at that moment," he said.
South Korea now meet the losers of Wednesday's semi-final between Turkey and Brazil in the third-place play-off in Daegu on Saturday.
"We have to swallow this disappointment," Hiddink said.
"Of course there's disappointment, but we'll recover. I hope to bring the players again to an acceptable level to go for third place.
"We must recover in three days for the public and for our own sake."
South Korea captain Hong Myung-bo, playing in his 131st international and at his fourth World Cup finals, echoed his coach's feelings.
"The German team looked really prepared," he said. "Thank you to the people for all the cheering and support."
Veteran striker Hwang Sun-hong thought the extra-time exertions against Spain were a factor.
"We were not able to recover fully. Germany has played more big matches and they were able operate better than us," he said.
"There are a lot of regrets but we tried our best. This is only the beginning for Korean soccer."
Midfielder Park Ji-sung was also in positive mood. "I believe we gained a lot from the tournament, all the players gained more confidence," he said.
"This is a big stepping stone for the further development of
Korean soccer. We will not be satisfied with just reaching the semi-finals."