Some 300 people had begun to gather for a demonstration backing Russia’s recent intervention in Syria when the rockets crashed into the embassy compound in the Mazraa neighbourhood of the capital, the journalists at the scene said.
There was widespread panic, but it was not immediately clear if anyone had been wounded or killed.
The demonstrators had been waving Russian flags and holding up large photographs of Russian President Vladimir Putin.
The Syrian Observatory for Human Rights monitoring group said the rockets were fired from the eastern edges of the capital, where Islamist rebels are entrenched.
The Russian embassy has been the target of rocket attacks before.
On September 21, just nine days before it began its air war in Syria, Moscow demanded “concrete action” after a shell hit the embassy’s compound in Damascus.
In May, one person was killed by mortar rounds that landed nearby. Three were hurt when mortar rounds landed inside the compound in April.
Al-Nusra calls Jihadists to attack Russia
The head of Al-Qaeda’s Syrian affiliate has called on jihadists in the Caucasus to attack Russian civilians and soldiers in retaliation for Moscow’s air strikes in Syria.
“If the Russian army kills the people of Syria, then kill their people. And if they kill our soldiers, then kill their soldiers. An eye for an eye,” Abu Mohamed al-Jolani, the head of Al-Nusra Front, said in an audio recording released late Monday.
He pledged that Moscow’s air war in Syria, which began on September 30, would have dire consequences for Russia.
“The war in Syria will make the Russians forget the horrors that they found in Afghanistan,” Jolani said, adding: “They will be shattered, with God’s permission, on Syria’s doorstep.”
Russia has said its aerial raids are targeting the Islamic State group, a jihadist rival of Al-Nusra, as well as other extremist groups.
Moscow’s strikes have concentrated on areas in northwestern Syria where Al-Nusra has a powerful presence, mostly in the province of Idlib.
A US-led air coalition fighting IS in Syria has also targeted Al-Nusra on several occasions.
Jolani called on armed opposition groups to set aside their differences until both air wars had been defeated.
“Delay the disputes until the demise and smashing of the Western Crusader and Russian campaign on Syrian land,” he said.