Also Sunday, an American foreign correspondent and his crew escaped unharmed during a firefight after being held captive for five days, said the reporter, Richard Engel of NBC.
The rebels fighting to topple Assad's regime have made significant advances in the past weeks, capturing air bases and military installations in and around Damascus.
Their offensive in the Palestinian refugee camp of Yarmouk in southern Damascus, which began Friday, is aimed at driving the pro-government Palestinian gunmen out of the camp.
When the revolt against Assad's rule began in March 2011, the half-million-strong Palestinian community in Syria stayed on the sidelines.
But as the civil war deepened, most Palestinians backed the rebels, while some groups - such as the Popular Front for the Liberation of Palestine-General Command - have been fighting on the government side.
By Tuesday afternoon, the Syrian military deployed several tanks along camp's main entrance, the residents said. There were no Syrian government troops in the camp and most of the fighting was between rebels and PFLP-GC gunmen, residents said. The group is led by Ahmed Jibril, Assad's longtime ally.
Activist videos posted online show armed men moving through the streets of the Damascus camp, as people cheer their presence and chant "God is great." Gunfire is heard in the background, and the narrator says the rebels are members of the Syrian Free Army.
The fighting in the camp has forced an exodus of Palestinian refugees and Syrians who came to the camp in past weeks to escape violence elsewhere in the city, according to United Nations officials. They told The Associated Press that the situation in Yarmouk is "chaotic."
Civilians continue to leave the camp, with some heading to UNRWA installations around Damascus. Others are fleeing to other Syrian cities and many are headed to the Lebanon-Syrian border, said Sami Mshasha, spokesman for the U.N. Relief and Works Agency.
A statement by UNRWA said the agency was "gravely concerned" for the Palestinian refugees in Yarmouk who "have experienced particularly intense armed engagements involving the use of heavy weapons and aircraft."
In Geneva, the United Nations said 4 million people need humanitarian aid in Syria.
World Food Program spokeswoman Elisabeth Byrs told reporters Tuesday that about 2.5 million Syrians need food aid, but her agency can only reach 1.5 million of them.
The World Health Organization said the Damascus Hospital gets 70 to 100 patients a day whose injuries mostly result from "the escalating violence over the last few months." Most of the injuries at one Damascus hospital are from burns, gunshots and injuries from explosions, said WHO spokesman Tarik Jasarevic.
The 21-month battle to bring down the Assad regime has forced some 3 million Syrians from their homes, according to a new estimate. Cold, wet winter weather is making life increasingly difficult for the displaced. Among those who left their homes are more than 500,000 who fled to neighboring countries - Turkey, Lebanon and Jordan.
After announcing that its chief foreign correspondent was free, NBC News said on its website that Engel and his team were kidnapped shortly after crossing the border from Turkey into Syria on Thursday. They were blindfolded and bound, but otherwise not physically harmed. The group was transported in the back of a truck to a location believed to be near the small town of Maarrat Misrin in northern Idlib province.
On Monday, the group was being driven to a new location when the kidnappers ran into a checkpoint run by members of the Ahrar al-Sham brigade, a Syrian rebel group. Two of the captors were killed in a firefight, the report said.
"We are pleased to report they are safely out of the country," NBC said in a statement.
The Syrian government has made it difficult for foreign journalists and citizens to report in Syria.
Those journalists whom the regime has allowed in are tightly controlled in their movements by Information Ministry minders. Other foreign journalists sneak into Syria illegally.
The kidnapping of foreigners has been rare, but as Syria descends further into chaos, the abduction of Syrians has become increasingly common.
In related developments, Russia's Foreign Ministry said Tuesday that two Russians were kidnapped along with an Italian in Syria, and their captors have demanded a ransom for their release.
The three, who worked at a Syrian steel plant, were kidnapped late Monday on the road between Tartus, where Russia has a naval base, and Homs, the ministry said in a statement. It did not specify the amount of the ransom demand.
Russia has been the key international ally of Assad, whose embattled regime is fighting rebel forces since March 2010 in a civil war that has killed an estimated 40,000 people.
On Tuesday, a Russian navy squadron set off for the Mediterranean, but the Defense Ministry did not say whether the navy ships are intended for an evacuation.
The squadron will rotate with navy ships that have been in the area since November, the ministry said.
Russia's base in the Syrian port of Tartus is its only naval outpost outside the former Soviet Union.
(Copyright 2012 by The Associated Press. All Rights Reserved.)