The 22-year-old has signed a long-term contract after completing a medical at the club's Melwood training ground.
Goal.com understands that Allen has trebled is salary to £45,000-a-week wages, along with a £2 million signing-on fee, after completing the £15m move.
Allen, who impressed under Brendan Rodgers at Swansea City last season, becomes Liverpool's second summer signing after Fabio Borini arrived from Roma last month.
"I feel incredible. Everyone knows the history of this club, it's a massive club, and I'm delighted to have joined," said Allen.
"The passion that people here have for football is something I share and I want to be part of that.
"I'm looking forward to being part of some great years ahead for Liverpool."
Allen has spent the previous three weeks for the Great Britain men football squad on Olympic duty.
And Rodgers is thrilled to have obtained the services of the Welshman as he continues to instil his tried-and-tested philosophy at Anfield.
Rodgers added: "I'm absolutely delighted that Joe has made the decision to come with us on this journey.
"Joe is a player whose profile will fit perfectly with the ideas of this group. His ability to control and dominate the ball is an important ingredient in our attempt to gain success on the field.
"Joe has had a fantastic education at Swansea City and will now begin the next chapter in his exciting career, and I wish him all the very best in what I'm sure will be a long and distinguished career here at Liverpool."