Barcelona can field virtually a reserve side when the teams play the return match in Valencia on 11 February.
“This was one of my most painful nights in football,” the former Manchester United and England full-back said.
“The scoreline is unacceptable. I won’t sleep well tonight. I didn’t like what I saw. The fans didn’t deserve that tonight. We have to recover incredibly quickly."
The 40-year-old told reporters after the game that he would not quit, despite a difficult start to his managerial career.
Since Neville's appointment in early December, Valencia have failed to win in eight league matches. Their only successes have come in the Copa del Rey. Following Wednesday night's mauling, glory in that competition is now a forlorn hope.
"Positivity has been immovable in my life,” Neville said. “I last had doubts as a player 18 years ago and from that moment on I developed a mechanism to deal with situations like this. Be clear, we have to deliver on Sunday against Real Betis.”
Valencia's sporting director Jesus Garcia Pitarch described the defeat as one of the worst results in the club's history. Pitarch said he had met club president Lay Hoon Chan after the match.
"We apologise to our fans for one of the worst days and one of the worst results in our history in a Cup semi-final," he told Canal + TV. "We know what we have to do. We know the situation the club is in is difficult. We have to reflect internally while recognising we have caused tremendous disappointment to our fans."