"A car suicide bomber blew himself up near a checkpoint at an entrance of Dafniya," between the town of Zliten and Libya's third city Misrata, said a spokesman for Fajr Libya.
The attack killed five fighters and wounded seven others, he added.
IS claimed responsibility for the attack in a message posted on Twitter, identifying the suicide bomber as a Tunisian named Abu Wahib al-Tunsi.
The jihadist group also warned Fajr Libya to be ready for "war".
"The apostates of Fajr Libya... must know that a war is coming to cleanse the land of their filth unless they repent and go back to their true religion," said the extremist group.
The acting prime minister of the Tripoli government, Mohammed Khalifa al-Guwail, responded by calling on armed forces to mobilise against IS in a statement read on television.
"The Tripoli government is determined to continue fighting extremism and criminal gangs who operate under what is known as the Islamic State until they are uprooted," the statement said.
It urged "officers, soldiers... and all security forces and revolutionaries to mobilise" against IS, describing it as a "takfiri" (Sunni radical) group that poses a "big threat" to Libya's security.
The Tripoli government also called on the international community to help it in its battle against IS by providing "technical, logistical and intelligence" support.
Libya plunged into chaos after the 2011 NATO-backed uprising that toppled and killed dictator Moamer Kadhafi, with battle-hardened and heavily armed former rebels carving out their own fiefdoms.
After Fajr Libya seized power in Tripoli last year, installing a parliament and government, the official administration fled to the east of the country.
- Oilfields at risk -
On Saturday, the recognised government also pleaded with the international community to provide it with weapons to battle IS.
A statement warned that IS plans on seizing "oilfields to fund its operations", as it has done in Syria and Iraq where the jihadists last year set up a "caliphate" straddling both countries.
On Thursday, IS seized control of the airport in Sirte, Kadhafi's hometown east of Tripoli, after Fajr Libya fighters withdrew from the facility.
It was the first time that IS in Libya has recorded such a military gain.
The Gardabiya airport, which is also a military base, lies just 150 kilometres (93 miles) from Libya's so-called "oil crescent" -- home to key oilfields and export terminals.
Officials in Tripoli said that IS had allied with supporters of the ousted Kadhafi regime to deploy across Sirte.
Sirte has been rocked by sporadic fighting between IS and Fajr Libya since February, when jihadists deployed in the city, capturing government buildings and the university.
The city, located 450 kilometres (280 miles) east of Tripoli, was already a bastion of Islamist extremist groups, including Ansar al-Sharia.
The radical group is classified by the United States as a "terrorist" organisation and suspected of involvement in the 2012 attack on the US consulate in the second city of Benghazi.
Tripoli officials say that IS, which also has positions in the eastern city of Derna, has "sleeping cells" in the capital, where the group has already claimed several attacks.
In a bid to find a solution to the crisis, the United Nations has for months struggled to broker a deal between warring parties through the creation of a national unity government.
On Saturday night a UN-sponsored meeting in neighbouring Tunisia of Libyan mayors and municipal representatives ended with a declaration calling for the "swift formation of a government of national accord".
Special envoy Bernardino Leon said on Thursday that the UN was preparing a new draft peace agreement which it plans to hand over to the rival factions in the first week of June.