The push into Hajar al-Aswad and Yarmouk came after government forces captured the last rebel-held southern and eastern suburbs of Damascus, boosting security in Assad's seat of power.
The month-long battles have been the toughest fought by the Syrian army and its allied forces this year against opposition forces in pockets around the capital, defense experts say.
The gains by President Bashar Assad's troops bring greater Damascus - including its far-flung suburbs - fully under government control for the first time since the war began in 2011.
Elsewhere, Kurdish forces are in control in much of Syria's east and northeast; while Turkey has sent forces to the northwest to fend off the Kurds.
The ultra-hardline group Islamic State, which was driven from most of the Euphrates River valley a year ago, now controls only two besieged desert areas in eastern Syria.
Islamic State group also captured a third of neighboring Iraq in 2014 but was largely defeated there past year.
The TV quoted an unnamed military official as saying a truce had been in place to evacuate women, children and elderly on Sunday night from Damascus' southern neighborhood of Hajar al-Aswad. State TV showed images of troops moving in, waving the Syrian flag and flashing victory signs atop wrecked buildings in the destroyed neighbourhood.
Around 1,000 fighters remained in the Yarmouk area at the time, of which at least 233 were killed in this month's fighting, according to the Syrian Observatory for Human Rights monitoring group.
The Syrian Observatory for Human Rights, a United Kingdom -based war monitor group, said there was a Russian-brokered deal between the Syrian government and IS militants to leave the area.
Syrian government forces launched an offensive on these areas on April 20 seeking to put an end to the presence of terrorist groups in southern Damascus and unblock the strategic Damascus-Amman highway.
In the southwest, where rebels hold territory at the Israeli and Jordanian border, Assad faces the risk of conflict with Israel, which wants his Iranian-backed allies kept well away from the frontier and has mounted air strikes in Syria.
The intervention in the conflict by Russian Federation and Iran, both staunch allies of Mr Assad, has turned the tide of the war in his favour in recent years and allowed him to re-establish control over the capital and other major cities in western Syria.
The Syrian government retaking the area surrounding Damascus and its countryside comes six years after losing its grip over large sections in the region and seven years since the armed clashes began in the area.
Gen. Ali Mayhoub, a Syrian army spokesman, declared Damascus and its surroundings "completely secure" on Monday.