Clashes erupted between the militants and groups inside the camp, with IS seizing control of large parts of the camp, reports said.
The UN says about 18,000 Palestinian refugees are inside the camp.
IS militants have seized large swathes of territory in eastern Syria and across northern and western Iraq.
But this is the group's first major attack near the heart of the Syrian capital.
Yarmouk residents told BBC Arabic that members of Aknaf Beit al-Maqdis, a group formed by Palestinian militiamen opposed to the Syrian government, were leading the fight against the IS militants, along with some Free Syrian Army fighters.
IS fighters had seized control of large parts of the camp, an official with the Palestine Liberation Organisation based in Damascus, and the UK-based Syrian Observatory for Human Rights, said.
There has been no official statement from IS about the move.
Unrwa, a UN relief agency for Palestinian refugees, said in a statement that the fighting would place Yarmouk's civilians, including large numbers of children, "at extreme risk of death, serious injury, trauma and displacement".
It demanded an end to the fighting and "a return to conditions that will enable its staff to support and assist Yarmouk's civilians".
Analysis: Lina Sinjab, BBC News, Beirut
Members of the self-proclaimed Islamic State stormed into the southern side of Yarmouk camp in the early hours of the morning and clashed with the Palestinian brigade, Aknaf Beit al-Maqdis.
Reports suggest they came in from the area of Hajar al-Aswad in the south of the capital.
The Palestinian Ambassador to Damascus, Anwar Abdulhadi, told the BBC that the group had seized the area of the camp near the Palestine Hospital.
Most information is coming from Palestinian officials in areas under government control.
The attack comes days before a deal to ease the humanitarian situation for civilians in the camp was set to come into operation.
Desperation in Yarmouk
Story of the Syrian conflict
What is Islamic State?
Syria's bloody conflict, which has entered its fifth year, has claimed the lives of more than 200,000 Syrians.
The battle between forces loyal to President Bashar al-Assad, rebels opposed to his rule and jihadist militants from Islamic State has also driven more than 11 million people from their homes.
The al-Qaeda linked al-Nusra Front captured the north-western city of Idlib from government forces on Saturday. It is only the second provincial centre to fall into rebel hands since IS seized Raqqa in March 2013.
Al-Nusra Front said on Wednesday that the city would now be ruled under Sharia, but that the group did not intend to monopolise power in the city.
Also on Wednesday, Jordan closed its only official border crossing with Syria due to fierce clashes between Syrian rebels and government troops.
Jordanian authorities said the Nasib border crossing, in the north of the country, had been closed as a preventative measure to protect travellers.
First built for Palestinians fleeing the 1948 Arab-Israeli war, Yarmouk was once considered by many to be the de facto capital of the Palestinian refugee diaspora.
Prior to the Syrian civil war, it had more than 150,000 refugees living there, and its own mosques, schools and public buildings.
However, the camp has been besieged by fighting between government troops and rebel forces since 2012.
Unrwa says about 18,000 refugees remain trapped in the camp, with inadequate access to food supplies, clean water and electricity.
In March, Unrwa said: "The extreme hardship faced by Palestine refugees in Yarmouk, but also in other locations in Syria as a result of the armed conflict is, from a human point of view, unacceptable."
Islamic State (IS) militants have entered the Palestinian refugee camp Yarmouk in Damascus, activists and Palestinian officials say.