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.
While it is very difficult to find out exactly what is happening in besieged eastern Aleppo, activists say it appears that after further fighting on Monday the rebels have lost all of the northern neighbourhoods they controlled before the weekend, leaving them with about two-thirds of the territory they had in the city.
Russia’s defence ministry says Syrian government troops have captured 12 districts, or 40 per cent of the territory, from the rebels.
Aerial bombardment of rebel-held areas was continuing on Monday, according to the monitors.
The Syrian army and its allies launched a major offensive to retake control of Aleppo in September.
Thousands of residents of east Aleppo have fled to areas controlled by government forces and Kurdish groups since the fighting intensified on Saturday.
The Syrian Observatory for Human Rights said as many as 10,000 residents fled to government-controlled western areas and a Kurdish-run northern district.