Allen completed a move to Anfield on Friday for a fee said to be in the region of £15m, a price Rodgers believes is a bargain.
Rodgers said: "When you see this boy play you'd think he was a Spanish player, a real European player. I would have paid many more millions for this kid.
"Joe is unique in that he's a British player who doesn't give the ball away. When he comes into this team, you will see the difference he can make.
"He wants the ball all the time. He has so much courage to get on the ball and play and I think he'll fit in really, really well with all the other clever players we already have here."
Liverpool play their opening Premier League game with Rodgers as boss against West Bromwich Albion on Saturday, August 18.