Renewed civil war in Libya splintered into areas of factional control since the 2011 overthrow of Muammar Gaddafi, threatens to disrupt oil and gas supplies, trigger more migration to Europe, and allow Islamist militants to exploit the chaos.
The death toll is higher than numbers reported by either side, said World Health Organization spokesman Tarik Jasarevic at a Geneva news briefing.
Ibrahim Fadel, an official at Mitiga, said no casualties were reported in the airport attack.
The airport has been closed down and all further flights suspended.
Libya is now split by two rival factions, with one supporting Haftar in the east, and another based in Tripoli in the west including a UN-backed government.
Seizing the capital, however, is a much bigger challenge.
The Libyan National Army carried out air strikes on southern Tripoli, while it seeks to advance towards the centre of the city from the abandoned airport.
Mismari said that the LNA had lost only two members in the fighting.
On Sunday, Marine Corps Gen. Thomas Waldhauser, the head of U.S. Africa Command said the United States has temporarily withdrawn some of its forces from Libya due to "security conditions on the ground".
In central Tripoli, while there were no signs yet of military and security vehicles or personnel on the streets, shops and cafes were closing earlier than usual in the evening and residents were apprehensive about the prospect of violence.More news: Iran responds to USA , labels CENTCOM terrorist
A Reuters correspondent in the city centre could hear gunfire in the distance southwards.
Al-Serraj, 59, who comes from a wealthy business family, has run the Tripoli government since 2016 as part of UN-brokered deal boycotted by Haftar.
The UN has said that 2,800 people have fled from the fighting near Libya's capital, with the Ministry of Public Health already revealing that at least 25 people had been killed and 80 wounded. "The United Nations is deeply concerned for the welfare of the civilian population in the ongoing violence and of the implications of the attack on the airport".
United Nations envoy Ghassan Salame met Serraj in his office in Tripoli on Monday to discuss "this critical and hard juncture", the UN's Libya mission said.
Haftar casts himself as a foe of extremism but is viewed by opponents as a new dictator in the mould of Gaddafi, whose 40 years rule saw torture, disappearances and assassinations.
Renewed war in Libya - splintered since Gaddafi's 2011 fall - threatens to disrupt oil and gas supplies, trigger more migration to Europe, and wreck United Nations hopes for an election. The media office of Hifter's army said 22 of their troops had been killed since Thursday.
Analysts say Mr Haftar has swelled his ranks with Salafist fighters and tribesmen as well as Chadians and Sudanese from over the southern borders, claims dismissed by the LNA.
Libya has become the main conduit for African migrants and refugees trying to reach Europe, with many detained if their journey fails and they are sent back.
France, which has close links to Mr Haftar, said it had no prior warning of his push for Tripoli, a diplomatic source said.
France established close relations with Haftar under the Socialist government of former President Francois Hollande and his defence minister, Jean-Yves Le Drian.