The Reds had to wait until the transfer window officially opened on 1 July in order to register the necessary paperwork and international clearance on the move, but that has now all been done.
Can is the third player through the doors at Anfield, with Southampton duo Rickie Lambert and Adam Lallana the other two.
"I am proud to be a part of the family and I am looking forward to playing here. I will give it my best," the 20-year-old told the club's official website.
"LFC is a great club and the fans are amongst the best in the world. I am looking forward to the first game and can't wait to step on the pitch for the first time at Anfield."
Manager Brendan Rodgers revealed he had been tracking the former Bayern Munich midfielder for some time before making his move a few weeks ago.
"Emre is an exciting young talent and I'm delighted he's chosen Liverpool for the next phase of his development and growth as a player," he said.
"He has recognised that we are a club that improves and nurtures exciting, ambitious players and I'm looking forward to seeing him take those next steps forward for us.
"We have tracked his progress for some time and I have been impressed with his attitude and qualities when I have seen him play both in the German Bundesliga and the Champions League.
"He has many of the attributes we look for; charisma on the football pitch and courage to want the ball and make things happen."