The advance "inflicted great human and material losses" on the militants, it said.
Amaq, a news agency affiliated to Islamic State, said its fighters had repelled the latest attacks on their positions in al Qadam and Hay al Tadamon, killing at least 17 army fighters killed in sniper attacks.
Regime forces have pounded southern districts of the capital since April 19 to try to expel the Islamic State group (IS) from the area, after the militants refused to leave under an evacuation deal.
Local sources said Wednesday that about 20 civilians have been killed by the Syrian regime bombing against Yarmouk refugee camp, which is mostly controlled by Daesh. The area is the last district out of government control in Damascus and its capture would boost security in Syrian President Bashar Assad's seat of power.
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The Yarmouk campaign is part of a wider Russian-backed offensive to regain the last pocket in opposition hands around the capital after retaking eastern Ghouta this month.
Regime forces were locked in violent clashes with ISIS fighters on Friday morning, the monitor said.
"The ongoing hostilities have caused deaths and injuries and the displacement of some 5,000 Palestinian refugees from Yarmouk camp to neighbouring Yalda area" the agency said, adding that there are "an uncertain number of civilians trapped in Yarmouk and urgently need a safe passage out of the camp".
Heavy air strikes and shelling had targeted Yarmuk and the edges of Hajjar al-Aswad and Qadam since the early morning.
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At least 85 regime personnel and 74 IS fighters have been killed in ten days of fighting in southern Damascus, the monitor said.
The militants have lost much of the territory they once controlled in both countries since they declared a cross-border caliphate there in 2014.
The United Nations Relief and Works Agency for Palestine Refugees (UNRWA) expresseD its deep recognition and gratitude for the extraordinary contribution of $50 million announced by King Salman Bin Abdulaziz Al Saud at the recent Arab League Summit in Dhahran, Saudi Arabia.
Before the outbreak of the Syrian civil war in 2011, Yarmouk was home to around 160,000 Palestinian refugees - people who had been displaced from their homes in modern-day Israel during the 1948 war, and their descendants.
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