The Serbia international rejected the opportunity to sign a new contract with his current employers in favour of entering a three-year deal with Frankfurt.
"We are delighted the transfer has been agreed," Frankfurt general manager Bruno Hubner told Deutsche Presseagentur.
"Despite his age, Aleksandar is already an experienced player able to shine in various positions. He is designed to play in the holding midfield role, but he could also be played in both full-back positions."
Bremen were also unsuccessful in their efforts to convince Aaron Hunt to sign a new contract with the club.