The ultra-hardline group, which has taken advantage of years of chaos to grab territory in the Opec member, said in an online statement it had also set off a suicide car bomb during the clashes, causing casualties.
There was no one from Libya’s authorities immediately available to comment on the reports of the fighting or the capture of Ben Jawad.
Es Sider and nearby Ras Lanuf oil ports, between Sirte and Benghazi on the Mediterranean coast, have been closed for more than a year amid fighting between rival factions for control of the North African state and its energy reserves.
Islamic State controls the city of Sirte and has attacked several oilfields in the south of Libya – though it has so far not taken control of any oil installations as it has done in Syria.
Libya has been split between rival governments, one based in Tripoli and the other in the east of the country, creating a security vacuum that militants have exploited.
Es Sider is protected by Ibrahim al-Jathran’s Petrol Facilities Guard, an armed faction which has backed the internationally recognised government in the east, but is also in conflict with other forces supporting that government.
The United Nations has been trying to win support for a deal brokered in Morocco last month to create a national unity government for Libya.
Separately on Monday, a military plane that was targeting militant groups in the eastern Libyan city of Benghazi was shot down, though the pilot ejected, senior army commander Fadel al-Hassi told Reuters.
Military forces allied to the internationally recognised government based in the east have been battling militants based in the city for months.