The Welshman joins Brendan Rodgers again, who also made the trip north earlier this summer to become coach at Anfield.
"You'd think he was a Spanish player, a real European player. I would have paid many more millions for this kid," said Rodgers.
"He is a unique player in that he's a British player who doesn't give the ball away. When he comes into this team you'll see the difference he can make.
"He's incredible on the ball, his intelligence for a 22-year-old is frightening, his game understanding is very, very good and he's in love with the football. He loves the ball.
"I think he'll fit in really well with all the other clever players we already have here. I'm excited about the prospect of Allen fitting in with our front four, because I think even at this early stage, their combination has been excellent."