Djourou came off after just 22 minutes during the 3-0 win, with an examination by Hamburg confirming that the defender now faces a lengthy spell on the sidelines.
The Arsenal loanee used Twitter to confirm that he is already on the comeback trail following tests at the training ground, vowing to return quickly as he looks to earn a permanent move to the Bundesliga.
He tweeted: "Hello. It is just a minor injury and I already feel better two days later.
"Any injury is always annoying but I'll come back strong."
The 26-year-old joined Hamburg on loan last summer after finding himself surplus to requirements at the Emirates, choosing to leave the club he joined in 2003 on a season-long deal.