The 22-year-old Serbian passed a medical yesterday and put pen to paper on a four-year contract with the German club on Wednesday.
He could make his debut for the German Bundesliga team in their home game against Cologne on Saturday.
Hamburg coach Michael Oenning admitted he could play immediately.
“I don’t want to rule anything out,” he told his club’s website.
“I have to get to know him a bit first. He is tall and an uncomfortable opponent. He has a quality that we don’t really have in our squad and that we need while he is left footed, which was particularly important to me.”
Due to work permit difficulties, the Serbia international spent practically all of his time as a Chelsea player out on loan in Holland.
He first joined PSV Eindhoven before spending two seasons with FC Twente, with whom he won the Dutch title in 2010, and then a year with Vitesse Arnheim, where he was made captain last season.
His form in the Eredivisie has also earned him a regular place in the senior Serbia national team and now he is ready to further in his career in one of Europe’s top leagues.
“Holland has a very good league for young players to develop in,” he said.
“The Bundesliga is the next step. It is a strong league, but I have played with PSV in the Champions League and Twente in the Europa League and so I hope to find my feet quickly in Germany.”
Having conceded 10 goals in their first three Bundesliga matches, Hamburg certainly needed to do something to shore up the leakiest defence in the league, which is where Rajkovic comes in.
“I am a central defender who is not scared of going in for tackles,” he added.
“I have only just arrived here so I don’t want to make any promises, but what I can promise is that I will always do my best and always give 100% for the team.”
Rajkovic links up with Eljero Elia, a team-mate from his Twente days, and a quartet of former Chelsea players – Gokhan Tore, Michael Mancienne, Jacopo Sala and Jeffrey Bruma – signed by the Bundesliga club this summer, under the auspices of their new director of sport Frank Arnesen.
Arnesen joined Hamburg in the summer after six years with Chelsea and he has used his links with the academy of his former club to account for five of the club’s six new arrivals this summer.