The rocket was fired on Thursday by militiamen from Fajr Libya (Libya Dawn), a coalition of Islamist fighters.
One oil 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.
Today the flames engulfed another two storage tanks at Al-Sidra, which is in the eastern region known as the "oil crescent" and home to other key terminals, he said.
"The mission warns of the environmental and economic consequences as a result of this violence and destruction in the oil crescent area, and urges the forces on the ground to cooperate in order to allow the fire crews to extinguish the blaze," it said.
Also Read
"They are doing their best to extinguish the fire and are working under difficult conditions," Hassi said.
The source, who declined to be named, estimated the amount of crude lost to the fire so far at more than 1.6 million barrels.
In its statement, UNSMIL called attacks on oil installations a "clear violation" of UN Security Council resolutions on Libya.
"Libyan oil belongs to all the Libyan people and is the country's economic lifeline," it said, urging all sides to "desist from any action that endangers this strategic national asset".