Elia joins Nils Petersen in departing the Green-Whites after the latter agreed a loan move within the Bundesliga to SC Freiburg.
"Elia has had his ups and downs here and now he's looking for a new challenge," said Werder sporting director Thomas Eichin. "We've tried to find a solution together and we've managed to come to an agreement that suits all parties."
The 27-year-old, who has won 28 caps for the Netherlands, scored five goals in 68 Bundesliga appearances for Werder since joining the club in 2012. The Dutchman has also previously played for Juventus, Hamburger SV and ADO Den Haag.