As the evacuation got under way, Syrian government forces and rebels battled in the suburbs of Damascus and elsewhere.
The Britain-based Syrian Observatory for Human Rights said clashes between opposition fighters and troops were concentrated in the areas around the capital, including along the road linking it to the international airport.
Persistent fighting along the airport road has prompted regional and international airlines to suspend flights to Damascus in recent weeks, although Syrian officials maintain that the airport remains open.
International diplomacy has done little to ease Syria's crisis, which according to the U.N. has claimed more than 60,000 lives since March 2011.
U.N. Secretary-General Ban Ki-moon said he didn't see "much prospect of resolution" by diplomacy.
At his first news conference of the year, Ban noted the worsening humanitarian conditions inside Syria and for those who have fled to neighboring countries.
For months, special envoy Lakhdar Brahimi, who represents the U.N. and the Arab League, has been pushing for a cease-fire and the formation of a transitional government — with little apparent progress.
Ban said he and Brahimi reached a bleak conclusion after talks Monday.
"Our shared assessment is that we are still a long way from getting the Syrians together," he said.
Associated Press writers Albert Aji in Damascus, Syria, Vladimir Isachenkov in Moscow and Bradley Klapper in Washington contributed to this report.