Real Madrid went out in the round of 16 after a dramatic 3-2 loss away to lower-tier opposition. Gonzalo Garcia thought he had rescued extra time with an equaliser in injury time, only for Jefte Betancor to score again in the 94th minute and seal a famous win for Albacete.
Speaking after the match, Arbeloa made it clear the blame lay with him.
"At this club, a draw is already bad enough, almost a tragedy. Imagine a defeat like this," Arbeloa said. "It hurts, and I know the fans are hurting too, especially when it happens against a team from a lower division."
He added: "If anyone is responsible for this defeat, it's me. I made the decisions about the line-up, how we wanted to play and the changes. I can only thank the players for the way they welcomed me and for the effort they made."
Arbeloa admitted the result was a harsh reminder of how demanding cup football can be for Real Madrid.
"We know how difficult these games can be. From here, we have to improve. My job now is to lift the players, physically and mentally, and focus on Saturday's game."
The former Madrid defender insisted the setback would not define him or his team.
"I'm not afraid of failure. I understand the criticism," he said. "Failure is part of the road to success. This defeat will make me better, and it will make us all better. We have a lot of room for improvement."
Asked whether the early exit could harm his chances of staying on as first-team coach, Arbeloa was firm.
"I've lost many times in my life. Defeats don't define you here," he said. "What defines this club are victories, titles and trophies. I'm here to help Real Madrid be the best in the world."
On what he would say to club president Florentino Perez, Arbeloa replied: "I don't have to explain anything to him. He knows this club better than anyone. Now it's time to work and make sure we respond in the right way."
Real Madrid's next match is against Levante UD on Saturday in La Liga. The game will be played at the Estadio Bernabeu.