The Western-proposed UN Security Council resolution, which would have covered war crimes committed by all sides in the civil war, comes less than a fortnight before President Bashar al-Assad's expected re-election.
Tanks and armoured vehicles rolled into the grounds of the prison in the main northern city of Aleppo, more than a year into a rebel siege of the sprawling complex, the Syrian Observatory for Human Rights said.
An army spokesman told state television this "tightens the noose around the terrorists' positions in the eastern and northern edges of Aleppo, and cuts off the roads (they) were using to move from the northern countryside towards the city."
The regime refers to the rebels as "terrorists."
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The advance into the area around the prison was preceded by an aerial assault today with highly destructive shrapnel-packed barrel bombs, said the Observatory.
Troops inside the prison have fought to defend it.
The prison had nearly 4,000 inmates before the siege, including Islamists. Poor humanitarian conditions, including a lack of food and medicines, and the bombardment have caused the death of some 600 prisoners, according to the Observatory.
State television showed footage shot inside the prison, and a reporter interviewed both security forces and prisoners, including women.
The latest success comes on the heels of another significant victory for the regime, which this month reclaimed the Old City of Homs, Syria's third city once dubbed the "capital of the revolution."
Millions of people are in desperate need of humanitarian assistance.
Despite the war, the Damascus regime is to stage a presidential election on June 3 in the areas under its control.