The exchanges of fire with heavy weapons killed at least six people and wounded 25, a health ministry official said.
It was not immediately known if civilians were among the casualties.
The assault on the Zintan militia which controls the airport by Islamist militants came after the UN pulled staff from Libya citing security reasons, and as Washington warned of further escalation.
An airport official said "rockets struck inside the airport perimeter around 6.00 am (local time)", followed by heavy clashes between the rival gunmen.
An airport source said Zintan fighters pushed back the assailants but that clashes continued to rage around the facility, as locals reported seeing tanks deploy and smoke billowing.
Authorities closed the airport for at least three days from today after initially halting flights.
The closure prevented Libya's foreign minister, Mohamed Abdelaziz, from travelling to a two-day meeting near Tunis opening today of his counterparts in North Africa to consider how to aid chaos-riddled Libya.
A foreign ministry source said Libya would be represented by its ambassador to Tunisia, Mohamed al-Maalul, at the closed-door meetings in Hammamet, a town south of Tunisia's capital.
The heavily armed group, named after a hill town southwest of the capital, is considered the armed wing of the liberal movement jockeying for power with Islamists who dominate parliament.
Today's attack was claimed by the Operations Cell of Libyan Revolutionaries, a coalition of Islamist militias seen as the armed wing of Islamists within the General National Congress (GNC) or parliament.
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