Mancini walked back to the dressing room at Doha's Education City Stadium after Abdulrahman Ghareeb missed a second Saudi penalty and handed match point to South Korea.
Wolverhampton Wanderers forward Hwang Hee-chan tucked it away to send the Koreans into a quarter-final against Australia, dumping Mancini's side out at the last 16.
"I apologise, I thought it was finished," the Italian said of his early exit.
"I didn't want to disrespect anyone," added Mancini, who won the European Championship with Italy in 2021.
"I want to say thank you to all my players for what they did. They are improving a lot."
Second-choice goalkeeper Jo Hyeon-woo was the hero for South Korea in the shootout, also saving from Sami Al-Najei to give his team a 4-2 win on penalties, after a 1-1 draw at the end of extra time.
Cho Gue-sung scored a 99th-minute equaliser in normal time to force extra time, after Abdullah Radif had opened the scoring for the Saudis 33 seconds after coming on as a half-time substitute.
Saudi Pro League sides have shelled out to lure big-name players while Saudi Arabia is set to host the 2034 World Cup, all part of a broader move into global sport.
Mancini took over as Saudi Arabia coach in August but he was unable to lead them to a first Asian Cup since 1996.
"I am very happy, I am very sad," he said.
"I am very happy because we improved a lot. We worked one month together and this was really important.
"Now we are a team. It's clear we have to improve more."
