Metropolitan Magistrate Ashok Kumar sent the Okhla MLA, who was produced in court, on police remand after Delhi Police sought his custody for investigation into the complaint.
Several protesters were seen shouting slogans outside the Saket court.
Khan, also the chairman of the Delhi Waqf board, was first detained for questioning and then arrested, a day after he alleged in a press conference that the woman was "pressurised" by police into giving a false statement against him.
According to police, on July 22, the woman had recorded her statement before the magistrate under Section 164 of CrPC.
She had alleged while she was returning from the legislator's residence, a vehicle tried mowing her down and Khan was sitting in it.
Prior to that, she had on July 19 filed a complaint with police alleging that at the AAP MLA's residence in Jamia Nagar, a youth had on July 10 abused her and threatened that she would be killed if she did not stop politicising the matter.
An FIR was subsequently registered in this regard under section 506 (criminal intidation) and 509 (word, gesture or act intended to insult the modesty of a woman) of IPC.
After her statement to the magistrate, Section 308 (attempt to commit culpable homicide) was added to the FIR, a police officer said.
The woman had on Saturday filed three police complaints alleging she was receiving threats from unidentified persons, following which she was provided security.
The woman, a resident of Jasola, had last week in a police complaint said she had telephoned Khan on July 10 and later went to his Batla House residence to raise the issue of power cuts with him.
She claimed the MLA did not meet her and while she was returning, a youth came out of the house, abused her and threatened that she would be killed if she did not stop politicising the matter.
Later, the FIR was registered against an unidentified youth under sections 506 and 509 of IPC at Jamia Nagar police station.
Khan had, however, said he did not even know if the woman came to his residence.
AAP had on Saturday rubbished the allegation levelled against the MLA, claiming Delhi Police had "pressurised" the woman into giving a false statement against Khan.
At a news conference, the party played a video, in which the woman is purportedly seen telling someone "that the SHO of the Jamia Nagar police station asked her to add rape and murder charges in the FIR".
"The woman is in touch with several BJP leaders and she was also taken to the magistrate by a lawyer affiliated to the party.
"The video also states that police asked her to file the complaint," Khan had on Saturday said.
Khan had further said he has written a complaint to DCP (South-east) of the Delhi Police to file an FIR against the SHO and the woman for "framing" him.