The Koreans were aiming to win the tournament for the first time since 1960 but they failed to muster a single shot on target against a Jordan side ranked 64 places below them in the world.
Tottenham star Son said his team had been nervous and "did not want to make a mistake".
"Very disappointing, devastated about this result," the 31-year-old skipper told beIN Sports.
"Jordan are having an amazing journey at this tournament. They deserve it, they fought to the end.
"For us it was a very disappointing night."
South Korea were on the brink of elimination in both previous knockout rounds but survived after scoring equalisers in second-half injury time.
This time they had no reply when Yazan Al-Naimat and Mousa Al-Tamari fired Jordan in front in the second half, with Jurgen Klinsmann's men rarely looking capable of scoring.
Jordan reached the final for the first time and will face either Iran or Qatar for the title on Saturday.
Son said South Korea had to "learn from these mistakes".
"We have to look forward, there is no time for regret," he said.
"Now I have to go back to my club and be ready for the rest of the season."
Son scored three goals at the tournament - two coming from the penalty spot and one a free-kick in extra time that gave them a quarter-final win over Australia.
The forward apologised to Korean fans, saying he was "terribly sorry that we didn't live up to their expectations".
"I will try to get better as a player and also help our national team take the next step," he told reporters.
