The attack was carried out by the Benghazi Defence Brigades, including fighters driven from Benghazi city by forces loyal to Haftar, the source said.
The fighting took place south of Ras Lanuf and Al-Sedra, two terminals severely damaged in clashes between the two groups in 2016 and 2017.
One of the reservoirs was hit by a rocket fired by the attackers, the same military source said.
A spokesman for Haftar's self-styled Libyan National Army, which controls most of the country's east, said the attack had been repelled.
"The attack aimed to reduce pressure on the terrorists in Derna," the LNA spokesman added.
The LNA launched an offensive on May 7 to "liberate" the coastal city of Derna, home to about 150,000 people and held by hardline Islamist fighters since the 2011 uprising that ousted and killed dictator Moamer Kadhafi.
The city is the only major part of eastern Libya outside the LNA's control. It is held by a coalition of jihadist militias, including groups close to Al-Qaeda, who oppose both Haftar and the Islamic State group.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)