There was also reported interest in the player from Premier League outfit Newcastle United, but Lasogga has now signed a five-year deal with the club he played for on loan last season.
"It's fantastic that this transfer has now been perfected and that my situation has been resolved," the 22-year-old told Sky Sports News.
"I'm delighted that I can continue playing for HSV. I hope this will be the start of a new and successful era and I'm pleased to be able to do my bit in the creation of a new Hamburg."
Lasogga scored 13 times in the league for Hamburg last season.