Rune, who won a tight match in five sets, was 2-1 up and 40-all in the third set when he clearly failed to get to the ball in time. But the second bounce was not picked up by chair umpire Kader Nouni despite being clearly visible.
Rune did not say anything and Cerundolo was furious with the decision.
"So when I was hitting the ball, I didn't know, I just ran for it," Rune said. "But then obviously when I saw it, after he did call it, I saw it after the next point on the TV, and I saw it was a double bounce.
"But the point already happened and he called the score. So I felt sorry. Sorry for him (Cerundolo). Then I managed to break him. I held serve. Then after he broke me it was close again."
A fuming Cerundolo, who told the umpire he would get him fined, eventually lost the set. He did battle back to win the fourth but Rune snatched the quarter-final spot with a 7-6(3) 3-6 6-4 1-6 7-6(7) victory.
"Some umpires make mistakes. Some for me, some for him. That's life," Rune said.
The Argentine called it a "huge mistake" by the umpire.
"It was a huge mistake. It was unbelievable with the double bounce," Cerundolo said. "The umpire has to check it again. It is tough to give the point away.
"It was tough moment. He (Rune) was not good not to give me to point but I understand it with the tension.
"They have to do something with the cameras because you cannot do that mistake. For sure I wish he (Rune) would have given it... if he saw it is bad for him. If he didn't see it and it was after, I understand."
But he said it did not hugely impact his performance after he battled back to win the fourth set.
"I didn't let that affect my game and situation," Cerundolo said.
Rune will next play Casper Ruud of Norway in a repeat of last year's quarter-final.
