The two clubs agreed a deal for the 19-year-old Dutchman earlier this month and the move was formally rubber-stamped as Bruma put pen to paper on a 12-month deal with an option for a further year.
"Hamburg is the best option for me," Bruma told Hamburg's official website.
"Here, I can play at the highest level in Germany."
Hamburg sporting director Frank Arnesen played a key role in luring Bruma to HSV, just a few weeks after taking up his new role after leaving a similar post at Stamford Bridge.
Bruma, who is under contract at Chelsea until 2014, will join up with former Blues team-mates Michael Mancienne, Jacopo Sala and Gokhan Tore, all of whom recently penned permanent deals at the Imtech Arena.