The 25-year-old joined HSV in 2009 and has 18 months remaining on his deal but has been deemed surplus to requirements.
But Fortuna general manager Wolf Werner has welcomed the player's arrival in Dusseldorf in the belief that his experience could prove vital as the club looks to move up the table.
"Robert Tesche is vastly experienced having played more than 100 top-flight matches," Werner told the club's official website.
"He knows the Bundesliga and is aware of the standards. I am sure he will be a useful player for us for the remaining months of the season."
Fortuna do not possess the option to sign the player permanently as part of the deal.