The Reds signed the Ivory Coast international on a free transfer from Manchester City in the summer to help fill the void created by Carragher's decision to hang up his boots.
The 32-year-old brought with him a wealth of experience having been a member of title-winning sides at Arsenal and City.
He made his Liverpool debut in last week's season-opening Premier League win over Stoke and Rodgers is pleased with the way he has fitted in.
Rodgers said: "He is a good man. He is funny to watch sometimes when he is running about, he has always got a smile on his face.
"One thing he has got is a winning mentality. He has got great belief in what we are doing here.
"His first leap against Peter Crouch just shows you the fitness and power he still has.
"When he goes up for corners he looks like he can score. I think we have got a player who is very hungry still to do well.
"He respects everything about this great club, it is a huge privilege for him to be here. He is here to perform and do well.
"He is certainly a character and offers great leadership on the field, which is primarily what we got with Carra."