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."
But Saudi Football Federation president Yasser Al-Misehal made clear his unhappiness in an interview with Saudi broadcaster SSC.
"The coach's exit is completely unacceptable and we will discuss with him why this happened," he said.
"He has the right to explain his point of view and then we will decide the appropriate action."
The president said that no decision had yet been made in regard to Mancini's future with the team, but the manager has every intention to remain in his position going forward.
"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."
