Rebels, who have driven President Bashar al Assad's troops from much of northern Syria and taken several crossing points to Turkey, overran another frontier town late on Thursday, a rebel commander and opposition sources said.
The fighting coincided with talks in Qatar aimed at creating a more representative and credible Syrian opposition body.
Ten people were killed in clashes as rebels took Ras al-Ain, an Arab and Kurd town in the northeastern oil-producing province of Hasaka, 375 miles from Damascus, the sources said.
"The crossing is important because it opens another line to Turkey, where we can send the wounded and get supplies," said Khaled al-Walid, a commander in the Raqqa Rebel Division.
The report could not be independently confirmed.
The UN High Commissioner for Refugees (UNHCR) said that in addition to the refugees fleeing to Turkey, thousands more had crossed into Lebanon and Jordan, also during the same 24-hour period, bringing the number of Syrian refugees in neighbouring countries to more than 400,000.
In the last three months, the mainly Sunni Muslim Arab rebels have captured outposts on the 560-mile border, steadily moving toward the north-east, home to many of Syria's one million Kurds. The encroachments have enraged several Kurdish groups ,which have tried to stay out of the violence.
The Kurdish Council, a coalition of Kurdish parties opposed to Mr Assad, called on the Free Syrian Army to leave Ain al-Arab, saying the clashes, as well as fear of Syrian army bombardment, had prompted most of the town's 50,000 inhabitants to flee.
"While the Kurdish Council affirms it is part of the revolution to bring down this totalitarian regime, the province of Hasaka must remain a safe area for thousands of refugees who had fled to it from other regions," it said.
Adrian Edwards, spokesman for the UNHCR, said there had been a large movement of Syrian refugees into Turkey's Urfa province, which borders Ras al-Ain, between Thursday and yesterday, a period coinciding with the rebel offensive.
The Turkish state-run Anatolian news agency reported that 26 Syrian military officers had also arrived in Turkey with their families overnight, in the biggest mass desertion of senior soldiers from Mr Assad's forces in months.
Turkey already shelters more than 120,000 Syrian refugees.
The Syrian Observatory for Human Rights said at least 12 people were killed by army shelling in the eastern town of al-Qurriya. The authorities denied responsibility.
Rebels and soldiers clashed in Kfar Souseh, a rich neighbourhood of Damascus a couple of miles from the presidential palace, activists said, adding that the area was being shelled from military bases in the capital.
On Wednesday, rebels fired mortars at the palace but missed.
An opposition activist in central Damascus said police stations and state buildings had been fortified with sandbags to guard against increasingly bold rebel attacks.
"First we saw the most important buildings protected. Now we are seeing police stations protected," said the activist, who asked not to be named for security reasons.