Rebel group Jaish al Islam said four of its fighters were killed trying to stop the army from cutting the Castello Road, the only route into rebel-held areas of Syria's second city.
Regime bombardment of several opposition-controlled districts left 10 civilians dead, including three children, the Syrian Observatory for Human Rights said.
Rebel fire on the government-held Sayf al Dawla neighbourhood meanwhile killed five people, including a child, the monitor said.
Russia planes pound Aleppo as Syria army closes in
The army announced on Wednesday that it was observing a 72-hour nationwide ceasefire for Eid. But fighting has continued, particularly in and around Aleppo.
President Bashar al Assad's forces have been trying to cut the Castello Road for more than two years and the Observatory said that Thursday's advance brought them the closest so far to achieving that goal.
Their capture of a hill just a kilometre away allows them to fire on any traffic moving along the road from the ground as well as from the air, the Britain-based monitoring group said.
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