Carrick said: "What can you say in the dressing room? Everyone knows we have let the club down.
"The fans turned out in their thousands and we all wanted a cup run. The gaffer made us fully aware of what the FA Cup means to this club before the game.
"The fans were magnificent, even when it was 2-2 they were singing about the cup and we are disappointed to let them down.
"I don’t think we took our foot off the pedal when we went 2-0 up. We just didn’t deal with a cross and let a goal in before half-time which gave Leicester a lift. At 2-2 we thought ‘OK we’ve got a replay’ but the last-minute goal has killed us.
"We had two or three players missing at Manchester City as well but managed to win that - so it’s no excuse. We just didn’t play well in the second half and paid the price. The cup’s the cup and we’re out of both of them."
Keane said: "The lads are all devastated. It is not nice to be out of the cup.
"To be knocked out in the third round is very disappointing. We had wanted to put on a big performance for the fans.
"We lost the goals through our own mistakes. Take nothing away from Leicester though, full credit to them, they played well and took the game to us."