A rare error from Carragher proved costly in the early stages of the contest as his under-hit back pass allowed Hulk to bear down on goal and slot beyond Pepe Reina for the opener.
Rodgers reflected: "It's probably the first mistake I have seen him make since I came to the football club - and that includes every day in training.
"There's no blame attached. I thought he was brilliant tonight in how he controlled Hulk, who is a real threat with his strength and power. It's just one of those unfortunate things a player.
"I'm disappointed for Carra because he's a brilliant professional and he deserved to go into the next round. Unfortunately, it wasn't to be."