The US is investigating reports that its air strike on Sunday killed 10 members of one family, ITV News Global Security Editor Rohit Kachroo reports
The rockets struck in and around Kabul's Salim Karwan neighbourhood, followed immediately by gunfire.
The Islamic State group claimed the attack, and there are no reports of casualties so far.
US officials say they've intercepted other rockets apparently fired towards Kabul Airport.
Witnesses, who spoke on condition of anonymity for fear of reprisals, said they heard the sound of three explosions and then saw a flash in the sky. People then fled after the blasts.
Some of the rockets struck residential apartment blocks, witnesses said.
In the Chahr-e-Shaheed neighborhood, the remains of a four-door sedan used by the attackers could be seen.
The car had what appeared to be six homemade rocket tubes mounted in place of back seats. IS and other militant groups routinely mount such tubes into vehicles in order to move them undetected.
“I was inside the house with my children and other family members, suddenly there were some blasts," said Jaiuddin Khan, who lives nearby. "We jumped into the house compound and lay on the ground.”
US military cargo planes continued their evacuations at the airport after the rocket fire.
Five rockets targeted the airport, said US Navy Captain Bill Urban, a spokesman for the American military’s Central Command.
A defensive weapon known by the acronym C-RAM — a Counter- Rocket, Artillery and Mortar System — targeted the rockets in a whirling hail of ammunition, he said.
The IS statement, carried by the group's Amaq media outlet, claimed the militants fired six rockets.
In Washington, the White House issued a statement saying officials briefed President Joe Biden on “the rocket attack at Hamid Karzai International Airport” in Kabul.
“The president was informed that operations continue uninterrupted at HKIA, and has reconfirmed his order that commanders redouble their efforts to prioritize doing whatever is necessary to protect our forces on the ground,” the statement said.
On Sunday, a US drone strike blew up a vehicle carrying “multiple suicide bombers” from Afghanistan’s Islamic State affiliate, ISIS-K, before they could attack the ongoing military evacuation at Kabul’s international airport, American officials said.
The US said the strike killed two IS members, but relatives of those killed in Sunday's attack dispute that account, saying it killed civilians who had nothing to do with ISIS-K.
Najibullah Ismailzada said his brother-in-law, Zemarai Ahmadi, 38, had just arrived home from his job working with a Korean charity. As he drove into the garage, his children came out to greet him, and that's when the missile struck.
“We lost 10 members of our family,” Mr Ismailzada said, including six children raging in age from 2 to 8.
He said the missile killed another relative, Naser Nejrabi, who was a soldier in the Afghan army and an interpreter for the US military in his mid-20s, along with two teenagers.
US military spokesman Bill Urban acknowledged the reports of civilian casualties and said the US is investigating these.
He said: “We would be deeply saddened by any potential loss of innocent life."
The US State Department released a statement on Sunday signed by around 100 countries, as well as NATO and the European Union, saying they had received “assurances” from the Taliban that people with travel documents would still be able to leave the country.
The Taliban have said they will allow normal travel after the US withdrawal is completed on Tuesday and they assume control of the airport.
However, Afghans remain fearful of the Taliban returning to the oppressive rule for which it was once known.
There have been sporadic reports of killings and other abuses in the sweep across the country.
Earlier this week, an Islamic State suicide attack outside the airport killed at least 169 Afghans and 13 US service members.
The US carried out a drone strike elsewhere in the country on Saturday that it said killed two members of the Islamic State's local affiliate in Afghanistan, which has battled the Taliban in the past.