An elite interior ministry unit had entered the Mithaq district and were clearing it Wednesday, the sources said, while counter-terrorism forces retook an industrial zone Tuesday.
The militants are using the city terrain to their advantage, concealing car bombs in narrow alleys, posting snipers on tall buildings with civilians on lower floors, and making underground tunnels and surface-level passageways between buildings.
"We were very afraid," said one Mithaq resident.
"A Daesh anti-aircraft weapon was positioned close to our house and was opening fire on helicopters. We could see a small number of Daesh fighters in the street carrying light and medium weapons. They were hit by planes."
Most of those fleeing are from the eastern districts but residents of the besieged west, still fully under the militants' control, are increasingly attempting to escape, scaling bridges bombed by the coalition and crossing the Tigris by boat.
Despite shortages of food and water, most Mosul residents had stayed in their homes rather than fleeing as many had expected before the offensive began in October.
The UN refugee agency has said 125,568 people have been displaced from Mosul, a city of about 1.5 million, and more than 13,000 of those have fled in the five days since the US-led coalition renewed an offensive that had stalled for weeks.
That represents an increase of nearly 50 percent in the number of people who fled every day from Mosul over the several weeks of relative calm that ended last weekend.
Twelve weeks into Iraq's largest military campaign since the US-led invasion of 2003, security forces have retaken about a quarter of Mosul.
"Finally we have been freed," a second Mithaq resident told Reuters by phone. "We feared fighting would be fierce, but it was easy compared with other areas. Daesh members fled without putting up strong resistance."
Counter-terrorism units pushed into eastern Mosul in October but regular army troops tasked with advancing from the north and south made slower progress and the operation stalled.
After redeploying forces, Iraqi forces have been advancing on three fronts towards the Tigris river that bisects Mosul, in a second phase of the offensive.
Victory in Mosul would probably spell the end for Daesh's self-styled caliphate but in recent days the militants have displayed the tactics to which they are likely to resort when they lose the city, with bomb attacks in Baghdad, and attacks on security forces in territory they have lost.