The attack was spearheaded by the Benghazi Defence Brigades, a coalition of mostly Islamist militias and armed groups, and was backed by tribal fighters and former petroleum installation guards.
They "advanced towards the oil crescent and are 15 kilometres (nine miles) south of the town of Nofliya," said Colonel Moftah al-Megarief, who heads an oil facilities guard controlled by Haftar.
Nofiliya is on the edge of the North African country's strategic oil crescent, home to the main export terminals of Al-Sidra, Brega, Ras Lanuf and Zuwaytina which Haftar's forces seized in September last year.
However, an officer on Haftar's staff told AFP that MiG-23 warplanes and Mi-35 attack helicopters bombarded a convoy of Benghazi Defence Brigades vehicles south of Nofliya.
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"Several vehicles were destroyed," said the source, who spoke on condition of anonymity.
Libya's oil-exporting region is bitterly contested between the country's internationally recognised and Tripoli-based Government of National Accord (GNA) and a rival administration in the east backed by Haftar.
In December, the Benghazi Defence Brigades carried out a similar attack on the oil crescent, but were repelled by Haftar's forces.
Megarief said today's assault was launched from Al-Jufra air base in southern Libya.
Rocked by chaos since the fall of dictator Moamer Kadhafi in 2011, Libya desperately needs to relaunch its oil exports, the backbone of its economy.
The GNA has struggled to impose its authority as it faces a multitude of battle-hardened fighters who took part in the uprising that ousted Kadhafi.