The Bundesliga club announced the transfer on its website Wednesday. No financial details were provided, but German media reports the fee was in the region of £2 million ($2.9 million).
''I've just arrived and don't want to make any big promises, but I can promise I will always give my best, always 100 percent, for the team,'' Rajkovic said.
Since former Chelsea official Frank Arnesen arrived in Hamburg to become director of sports, the Bundesliga club has signed five young players from his former club.
The other four are Jeffrey Bruma, Michael Mancienne, Gokhan Tore and Jacopo Sala.
Hamburg coach Michael Oenning said the 1.91-meter (6-foot-3) Rajkovic, 22, could play when Hamburg hosts Cologne on Saturday in a match between the Bundesliga's bottom sides.
''He is big and an awkward opponent who steps in at the right moment. He can also spread fear and terror. He brings a quality that we don't yet have in the squad and that we need,'' Oenning said.
Hamburg has conceded nine goals in its opening three league games, including five to Bayern Munich last weekend.
Work permit issues prevented Rajkovic playing in the Premier League for Chelsea following his move from OFK Beograd as a 17-year-old. The Belgrade-born defender was loaned out to three Dutch clubs, PSV Eindhoven, FC Twente and Vitesse Arnhem.