Lasogga has penned a five-year deal after spending the previous season on loan at the Imtech Arena.
The 22-year-old scored 15 goals for Hamburg last season, hitting the target in their 1-1 draw with Greuther Furth in the second leg of the relegation play-off that ensured survival for Mirko Slomka's men on away goals.
Wolfsburg and Premier League side Newcastle United had been linked with a move for Lasogga.
However, it is Hamburg who have secured his signature.
"It's fantastic that this transfer has now been perfected and that my situation has been resolved," Lasogga told Hamburg's official website.
"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."
