The refugees came via the Tanaf border crossing 400km south of al-Sada, said Ali Moushawah, a lawyer in al-Boukamal, another border crossing between Iraq and Syria that is 12km from Sadah.
Moushawah said the refugees were mostly tribesmen who sensed that the US attack was coming.
They fled to join their kinsmen on the Syrian side of the border, he said.
"They used Iraqi passports to enter Syria at Tanaf because US forces had closed the al-Boukamal border checkpoint," he said.
The US air and land attack at al-Sada in the al-Qaim area began early on Saturday and was continuing on Sunday. The US military said at least eight fighters were killed in the gunbattles and an air strike in the first day of the offensive.
Moushawah said the roar of attacking aircraft and thuds of artillery shells intensified during the night and kept residents of Syrian border villages awake.
None of the villages was hit, he said.