German prosecutor Detlev Mehlis, accompanied by a security guard and a female aide, arrived at Mr Annan's New York residence shortly after 5:00 pm local time (2200 GMT).
The UN secretary general and Mr Mehlis shook hands for the cameras at a photo session. They made no statement to the press.
Ahead of the handover, Mr Mehlis expressed support for Lebanon's request to renew the mandate of the UN enquiry panel for another six months, saying his team had made progress but that the February assassination still required further investigation.
"We have so far reconstituted half the puzzle," Mr Mehlis told a Lebanese newspaper, Al-Mostaqbal, in an interview published on Saturday.
"There are still dark corners that we are shedding light on, but the enquiry is progressing," Mr Mehlis told the paper, which is owned by Mr Hariri's family.
UN diplomats said the Security Council was likely to pass a new resolution endorsing Lebanon's extension request after hearing a presentation from Mr Mehlis scheduled for Tuesday.
Members of the 15-member council will also receive copies of the Mehlis report "to give them about a day to study it," the UN said.
Mr Mehlis has said he would step down when his six-month mandate expires on Thursday to return to his work as a magistrate in Germany.
He is expected to give a press conference on Tuesday after briefing the Security Council.
In his interim report released in October, Mr Mehlis cited "evidence" that senior Syrian and Lebanese security officials were involved in Mr Hariri's murder on February 14 in Beirut.