A late Inaki Williams goal earned the victory for Bilbao.
"A fateful ending after all that happened," he said. "That's football. One cross has ended up in the back of the net. We'd managed the game well, we knew how to suffer but one detail and you're gone.
"The image of the team was good and we deserved to go through, but this is the new Copa. In one isolated moment, it can slip away from you.
Asked about what's been a difficult week at the club, with Lionel Messi's comments about Eric Abidal's interview, Busquets added: "We are just focused on the sporting side of things. We won in the league at the weekend but it wasn't to be today.
"It's a sad day for everyone. We didn't want to go out. We have two competitions left, we have to keep on fighting. We have to continue. Everyone's angry. We did things well."