The rocket was fired on Thursday by militiamen from Fajr Libya (Libya Dawn), a coalition of Islamist fighters.
One tank was hit, said the region's security spokesman Ali al-Hassi, before the fire spread on Friday to two other full tanks at Al-Sidra terminal.
Witnesses said huge columns of smoke filled the sky over Al-Sidra and were being blown further east towards Ras Lanuf terminal.
They expressed concern that the fire could spread to other tanks at Al-Sidra and cause an environmental disaster if not brought under control.
National Oil Company spokesman Mohamed al-Harari urged everyone to respect the neutrality of oil terminals, adding that the tanks that were hit were full and ready for export.
On Thursday, the Islamists killed 22 soldiers in a surprise attack during which they used speedboats in a failed bid to seize Al-Sidra and other terminals in the eastern region of Libya known as the "oil crescent".
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Hassi said the army repelled the Islamists.
On Friday, the fighting had subsided but reports said the situation was still tense.
The latest clashes pushed oil prices higher in Asia on Friday, with US benchmark West Texas Intermediate for February delivery rising 28 cents to $56.12, while Brent for February gained 13 cents to $60.37.
Military and medical sources said 18 soldiers and a Fajr Libya fighter were killed in Sirte, and another four soldiers were slain in Al-Sidra.
Most of the dead soldiers belonged to the 136th battalion responsible for monitoring a power plant west of Sirte, the sources said.
Since the clashes erupted on December 13, Libyan oil production has fallen to nearly 350,000 barrels per day compared with 800,000 previously, according to industry experts.
More than three years after dictator Moamer Kadhafi was toppled and killed in a NATO-backed revolt, Libya is still awash with weapons and powerful militias, and has rival parliaments as well as governments.
The Islamists say they have been mandated by the rival parliament based in Tripoli to "liberate" Libya's oil terminals.