"Faced with the advance of Islamic State, 400 families, about 2,000 people, were able to leave the camp on Friday and Saturday, from two secure routes to the adjacent Zahira neighborhood, which is held by the Syrian army and where shelters have been erected," Anwar Abdel Hadi told AFP.
The Britain-based Syrian Observatory for Human Rights and al-Dimashqi said those who fled the camp have reached the southern Damascus suburbs of Yalda, Babila and Beit Sahem, which are under rebel control.
There are still 18,000 people in the camp, a mixture of Palestinians and Syrians, according to the UN Relief and Works Agency (UNRWA). Yarmouk became a battleground before the siege, and has been devastated by street fighting, air attacks and shelling.
Some civilians were also evacuated from the camp on Sunday, the United Nations told Reuters.
Palestinian President Mahmoud Abbas said in Ramallah that residents of Yarmouk have been victimized by Syria's civil war. Government forces and different rebel factions are clashing and "we pay the price," he said.
He added that the Palestine Liberation Organization in Damascus has formed a "cell to handle this tragedy and they are trying to work it out with the least losses." Abbas said: "We are in touch with our brothers there to find a way out and protect our people."
The Syrian Observatory for Human Rights, which monitors the conflict from Britain, said the fighting has killed 26 people since the clashes first broke out.
In Damascus, Anwar Raja, the spokesman for Damsascus-based Popular Front for the Liberation of Palestine - General Command, which supports embattled Syrian President Bashar Assad, said several pro-Assad factions have united to defend the camp. He said more than 100 civilians have been either killed or kidnapped by the IS fighters whom he said now control about half the camp, adding that the priority now is to evacuate civilians.
Islamic State has taken most of Yarmouk from rival insurgents while the army surrounds the district, only a few kilometres from Syrian President Bashar Assad's palace.
UNRWA spokesman Chris Gunness said 94 civilians, including 43 women and 20 children, made it out of the camp on Sunday and were provided with humanitarian support.
The Observatory said hundreds had fled over the past two days.
"There have been some civilians who were able to flee the camp earlier today and we call on all parties to show maximum restraint so other civilians who wish to leave can be evacuated," Gunness said in an email.
The Observatory said the Syrian air force dropped crudely-made barrel bombs on Yarmouk on Sunday. Syrian state media did not report that civilians had been evacuated.
Saeb Erekat, from the PLO executive committee, called Saturday for the an evacuation of civilians. "Reports of kidnappings, beheadings and mass killings are coming out from Yarmouk," he said.
"For over 700 days, the camp has been the victim of a draconian siege, which has resulted in the death by starvation of at least 200 Palestinians. The PLO, through its envoys, has been trying for years to lift the siege."
Islamic State rules swathes of eastern Syria and Iraq and is the target of a US-led campaign of air strikes.
The group posted photos on Sunday of its fighters inside the camp. It also showed a photo of 13 men kneeling and facing a wall. The caption described them as rival fighters from Aknaf Beit al-Maqdis, an anti-Assad militia of Syrians and Palestinians from the camp.
Yarmouk was home to half a million Palestinians before the Syrian conflict began in 2011. The war has killed 220,000 people and displaced millions.