"I thought we were outstanding tonight," said Rodgers. "We were terrific for 44 minutes and 55 seconds in the first half but we just switched off at the crucial moment.
"We created chances throughout and although they got at us a bit more as we pushed on I thought we deserved something out of the game.
"I am very proud of my young players. It was a difficult night but it was a great team performance."
The defeat has seen Anzhi overtake Liverpool at the top of Group A with two games left to play.