The 20-year-old Can follows England World Cup pair Rickie Lambert and Adam Lallana in joining the Anfield club, who finished second to Manchester City last season.
"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," Rodgers told the club website.
"He has recognised 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.
"He has many of the attributes we look for; charisma on the football pitch and courage to want the ball and make things happen."
Can, who was born in Frankfurt and has Turkish ancestry, scored three goals in 29 Bundesliga appearances last term and also played seven times in the Champions League, having moved from German champions Bayern Munich.
"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 Bundesliga and the Champions League," Rodgers added.
Can, who played under former Liverpool defender Sami Hyypia before the Finn was sacked by Leverkusen last season, is able to take up several roles but wants to establish himself as a playmaker.
"It definitely is an advantage to play a number of positions because I can help the team in many different ways," he said.
"However, I have stated this before and I would like to mention it again, I see myself as a central midfielder, that is where I feel most comfortable. But, at the end of the day, the manager knows best where to play me."
(Reporting by Josh Reich, editing by Tony Jimenez)
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