More than half the storage tanks at both terminals were badly damaged in previous fighting and have yet to be repaired, though there have been regular loadings from Es Sider.
Libya's National Oil Corporation (NOC) said it had evacuated staff from the two terminals "for their safety". The production loss was around 240,000 bpd and the entry of a tanker due at Es Sider on Thursday was postponed, it said.
A military source said the three-pronged attack was launched by the Benghazi Defence Brigades (BDB), a group that has previously tried to take the oil crescent and advance on Benghazi, which has been fully controlled by Haftar since late last year.
Ibrahim Jathran, who headed an armed group that blockaded the terminals for more than two years before being forced out by the LNA, appeared in a video posted on social media announcing the start of a campaign.
"We announce the preparation of our ground forces and supporting forces in the oil region, and our objective is to overturn the injustice for our people over the past two years," he said, standing in a camouflage jacket in an unidentified desert area.
"The past two years have been catastrophic for people in the oil crescent because of the presence of the system of injustice which is the other face of terrorism and extremism."
Repeated previous attempts by the LNA's opponents to retake the oil crescent have failed, and it is unclear how much military and local, tribal support Jathran or BDB forces currently have.
However the LNA, which is the dominant force in eastern Libya and rejects an internationally recognised government in the capital, Tripoli, has been stretched thin.
Since last month it has been waging a campaign to take control of Derna, the last city in the east to elude its control.
Thursday's clashes were not affecting any oilfields, the military source said. The LNA had suffered casualties, he said, without giving details.
A local resident said he had heard the sound of heavy clashes and air strikes at dawn and had seen a large fire at the Ras Lanuf tank farm.
Crude exports from Ras Lanuf stood at 110,000 bpd in May, while exports from Es Sider were around 300,000 bpd, according to oil analytics company Vortexa.
The Minerva Lisa oil tanker, which was due to arrive at Es Sider to load a crude cargo on Thursday, was advised to stay outside the port, a source familiar with the matter said.
The tanker, chartered by trader Petraco, was seen turning away from the port on Thursday morning without loading, according to Reuters ship tracking.
A second tanker, the Seascout, is expected to arrive at the port on June 18.
Libya's oil production recovered last year to just over 1 million bpd and has been mostly stable, though it remains vulnerable to shutdowns and blockades at oil facilities.
National output is still well under the more than 1.6 million bpd Libya was producing before a 2011 uprising led to political fragmentation and armed conflict.
(Reporting by Ayman al-Warfalli, Ahmad Ghaddar and Aidan Lewis; Writing by Aidan Lewis; Editing by Mark Potter and David Evans) ((Aidan.Lewis@tr.com; +216-29850352;))