The advance, after heavy bombing from the air, is a major blow for the armed opponents of President Bashar al-Assad.
State TV said government troops were dismantling mines and explosives and continuing their advance.
Thousands of civilians have fled the besieged districts after a weekend of heavy fighting. Hundreds of families have been displaced within the area.
Rebel fighters have been driven out of more neighbourhoods of their long-held enclave in eastern Aleppo, as Syrian government forces continue to advance.
Both state TV and the UK-based monitoring group, the Syrian Observatory for Human Rights, said that the district of Sakhour had fallen to the Syrian army.
This would cut through the middle of the previously rebel-held territory, dividing it into two.
Thousands of residents of east Aleppo have fled to areas controlled by government forces and Kurdish groups since the fighting intensified on Saturday.
This advance follows two weeks of relentless aerial bombardment, as the government and its allies renewed their assault on the rebel-held portion of Aleppo.
It followed a three-week pause in the Russian and Syrian bombing of the area, which ended on 15 November.
Retaking the whole of Aleppo, Syria's second largest city, is a key aim of the Syrian government. The east of Aleppo has been held by rebel factions opposed to President Bashar al-Assad for the past four years.
The loss of eastern Aleppo would be a devastating blow for the rebels.
The tide is clearly turning in Aleppo in favour of the Assad regime - raising serious questions about the ability of the disparate rebel groupings to maintain a significant area of control in northern Syria.
The recapture of the city would be an important symbolic achievement for President Assad - underlining the fact that despite all the calls for his departure, Russian and Iranian support has been able not just to maintain him in power, but has enabled the regime to go onto the offensive.