The match culminated in a penalty shoot-out after neither team was able to make a breakthrough in normal and extra time.
It was a 50-50 game and the tension was at a super high level
South Korea scored all five of their penalties, but Joaquin missed for Spain and Hiddink's team will now meet Germany for a place in the final.
"I cannot describe how I am feeling right now, but above all I am so happy for the boys," Hiddink said.
"More dreams have come true. We'll have a little glass of champagne now and what comes tomorrow comes tomorrow.
"The mentality of this team to recover when getting ready to face tougher and tougher opposition is outrageous - it seems to be almost without limits.
"The fans have been fantastic, and all without any violence."
Hiddink felt that there had been little to chose between the two teams.
"It was a 50-50 game and the tension was at a super high level. And at the end we took our penalties very well.
"Spain have several experienced players and one or two times in the match they had superiority with their fast counter attacks. But then again we had periods like that as well."
But he was unhappy with suggestions that the officials in the match were biased toward his team.
"That is not fair," he said. "Errors are made, by players, by coaches and by referees and linesmen.
"The losing team should look in the mirror, not to external circumstances."
Hiddink looked ahead to his side's semi-final match against Germany.
"We approach the next game against the Germans like a bunch of young dogs, we have nothing to lose," he said.
"We have less recovery time than the Germans and so the schedule is not working for us."
He refused to be drawn on whether he will continue with his role after the World Cup.
South Korean president Kim Dae-Jung said the victory was the finest moment for the country since Dangung founded the Korean nation 5,000 years ago.
"This is the happiest day since Dangung and now a new path to national prosperity has been opened," he said.
"Now let's go to the final and win the tournament in Yokohama, let's make the World Cup ours."