The 27-year-old spent five years at Leicester, scoring 29 goals and providing 18 assists in 165 appearances after moving to the King Power Stadium from Red Bull Salzburg, but became a free agent when his deal was not renewed following the club's relegation to League One.
He has now put pen to paper on a contract with HSV running until 2027, with the option of a further year.
HSV sporting director Kathleen Krüger said: "With the signing of Patson we have strengthened our attacking position in a targeted way. With him we get a forward who very much matches the profile we have defined: he brings a huge work-rate both with and without the ball and consistently makes himself available for the team as a team player.
"At the same time, he has a great deal of experience at international level. We are convinced that Patson can help our team immediately with his qualities, and we are very much looking forward to welcoming him to HSV."
The move addresses a pressing need in Hamburg's squad after they managed only 40 Bundesliga goals last season, with no individual player reaching double figures. The club have already lost their two leading scorers from last term, Ransford-Yeboah Königsdörffer and Robert Glatzel, while loan forwards Philip Otele and Damion Downs have also returned to their parent clubs at Basel and Southampton respectively.
Daka will link up with a number of familiar faces from Scandinavian and African football at Volksparkstadion, including Yussuf Poulsen and Albert Grønbæk, as HSV look to build a more potent attack ahead of their second season back in the top flight following promotion.
The Zambian originally arrived in English football as one of the Premier League's more eye-catching signings, joining Leicester from Red Bull Salzburg for a reported €30m in 2021 after a prolific spell in Austria in which he helped the club win four successive league titles.
He struggled to fully establish himself as a regular starter in the Premier League and was part of the squad through a turbulent few years for the club, which included relegations from the top flight before this year's further drop into League One brought his time to an end.
HSV sporting director Claus Costa added that Daka combines energy, pace and intensity in his pressing, with the club hoping his experience translates quickly into goals.