Fighters for the Government of National Accord (GNA) said they had advanced into several districts and overrun a military camp. The LNA said it had already pulled out of those areas in what it said was a humanitarian gesture for Ramadan.
The GNA has with Turkish help made sudden strides in recent weeks, seizing a string of towns from the LNA, capturing the strategically important Watiya airbase and destroying several of its Russian-made air defence systems.
The LNA, backed by the United Arab Emirates, Russia and Egypt, has promised to respond with a massive air campaign and on Thursday its spokesman Ahmed Mismari said four warplanes had become newly available.
As diplomats warned of the risk of a new round of escalation with the warring sides' external backers pouring in new weaponry, the focus of the conflict shifted towards Tarhouna, the biggest remaining LNA stronghold in northwest Libya.
The GNA took Asaba, which lies on a major supply route to Tarhouna on Thursday, putting more pressure on LNA forces in the town, which has been subjected to bombardment for weeks.
An LNA military source said GNA forces were gathering to attack Tarhouna and said it had downed a drone there.
The United Nations Libya mission said it was following the mobilisation around Tarhouna with "great concern", warning all parties against any acts of retribution in a statement the United States said it supported.