The nine-times European champions won the second leg 1-0 at home in front of around 80,000 fans but still lost 4-1 on aggregate to the tiny Segunda B team, who play in the same division as Real's youth side, in the last 32.
"I understand the mood of the fans and their emotions but the club cannot act on emotions," Valdano told Spanish TV after some supporters had chanted for Pellegrini to quit.
"We remain calm. There is absolute confidence in the coach. We know we have to keep working intensively and with humility so we don't suffer any more surprises of this nature."
Chants of "Pellegrini quit" rang out around the Bernabeu for around 20 seconds when the Chilean took off midfielder Lassana Diarra in the second half and replaced him with defender Marcelo.
DIARRA WORRY
The coach dodged questions about his future and said he took Diarra off because the player had missed a week's training recently for family reasons and he did not want to risk him getting injured.
"Of course being knocked out of the King's Cup by a side from Segunda B cannot be considered a success," Pellegrini told reporters.
"We are in the middle of a campaign I am sure will end on a successful note, whatever has happened here today. I am thinking more about the team than myself."
Earlier on Tuesday president Florentino Perez played down the club's trophy expectations this season.
"Our dream is to build a spectacular team but it would not be a failure if we didn't win a title," Perez told Spanish television station Cuatro.
"We are at the start of a new project. We are giving it stability. We aren't going to get nervous because it hasn't all come together perfectly in the first year."
Perez's return in June for a second spell as president set in motion a 250 million euro reconstruction programme that brought in players like Cristiano Ronaldo, Kaka and Karim Benzema.
