The refugees were fleeing the Qalamoun mountains north of Damascus, a strategic border district with a mixed Sunni Muslim and Christian population where both sides have been massing forces.
"Some 1,000 families have fled to Arsal since yesterday," said Ahmad al-Hojairi, a local official in the Sunni Muslim Lebanese border town.
"We are trying to accommodate them in people's homes and in tents, but there is no way we can cover for their needs," he told AFP.
The families were crossing the border by car, by motorbike or on foot, he said.
"We expect more to arrive in the coming days as the fighting in Qalamoun intensifies."
Lebanon already hosts more than 814,000 Syrian refugees and is ill-placed to cope with a new influx fleeing the conflict in its larger neighbour.
But Syrian Observatory for Human Rights director Rami Abdel Rahman warned that Qalamoun area was "witnessing the lead-up to a major battle".
Both the Syrian army, backed by fighters of Lebanese Shiite militant group Hezbollah, and rebels, including Al-Qaeda loyalists, had sent reinforcements to the area, he said.
Residents on the Lebanese side of the border said they heard the sound of heavy shelling from the Syrian side throughout yesterday.
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