Despite leading 1-0 from the first leg, Real were eliminated on away goals as Leganes recorded a famous 2-1 triumph at the Bernabeu on Wednesday night.
Real are also realistically out of the running for the La Liga title, with Carvajal suggesting that the club must focus their attention on trying to defend their Champions League crown.
The 26-year-old is quoted by Marca as saying: "We're gutted, it's a painful situation as we haven't even been able to draw with Leganes at home. They used their weapons and they took advantage of our mistakes to score twice.
"We had chances to draw, but it didn't go our way. I think the team tried to win the game throughout the 90 minutes."Now, the focus is on hard work.
"We are in a difficult moment, it's the time to work harder than ever and be together. We will try to get back to winning ways in the only competition that we have."