The attack started shortly after 11am and at least two explosions were heard at about 2.30pm followed by a much bigger one about 15 minutes.
Reports indicated that anywhere between 30 and 40 explosions had been heard in the vicinity over a three hour period.
Airport officials have said all flights into Kabul have been diverted and flights out of the city have been cancelled until further notice.
However, the officials said no damage has been caused to the runway but that it is being cleared of shrapnel.
The ministry of interior confirmed that at least one civilian has been killed.
The ministry’s spokesman Najibn Danish said the rockets were fired from two locations, one in Qala-e-Mir Abbas area of PD9 and the other in Paktia Kot area, also in PD9.
Crisis Response Units have reportedly surrounded one of the locations where at least three insurgents are believed to be hiding.
“One of the rockets was launched in Paktia Kot area in a civilian house which killed one civilian and injured four others,” Danish said.
“The special forces have surrounded the house where three insurgents are hiding.”
Danish said insurgents are using RPGs and mortars, he said.
However, he did not provide further details.
This attack was launched just hours after US Secretary of Defense James Mattis and NATO chief Jens Stoltenberg arrived in Kabul on Wednesday.
The Taliban claimed responsibility for the attack and said the target had been Mattis.