Allen followed his manager to Anfield during the summer.
Rodgers said: "I thought he did very well. He was playing more as the controller. I thought he played well today. His body work was good, he received the ball well and played the ball forward at the right times.
"Joe has been terrific since he came in. The last few weeks obviously there have been things going on and becoming a father and no sleep, but across the board he's been brilliant."