The Reds had to wait until the summer transfer window officially opened on July 1 in order to register the necessary paperwork and international clearance on the £10m move, but that has now all been completed.
Can is the fourth new arrival at Anfield this summer, joining Southampton duo Rickie Lambert, Adam Lallana and former Tottenham defender Kevin Stewart.
The Germany U21 international joined Leverkusen from Bayern Munich last summer on a four-year deal and played in 29 Bundesliga matches, scoring three goals, and also featured in seven UEFA Champions League games.
"I am proud to be a part of the family and I am looking forward to playing here. I will give it my best,” Can told Liverpool’s 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 Can 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."