Reports had suggested that Carroll was left out of the squad due to a pending return to former club Newcastle United, but Rodgers denied this after the game.
"Andy was due to start," Rodgers told The Express. "He had a slight twinge on his hamstring. We had a scan on it and it didn't show a tear, just a slight disruption.
"We tried to give him 48 hours, let him travel and he trained to see how it was. It was still a bit tight and the medics' advice was it was too much of a risk for him to play.
"It was unfortunate because he has worked well and I know he was desperate to play. Hopefully missing the game will allow him to recover for the weekend."
Carroll has scored just six league goals for Liverpool in 43 appearances.