The footage shows members of the Syrian Civil Defence, also known as the White Helmets, entering the ruins of a house in the city's Tishreen neighbourhood.
A military media unit run by Damascus ally Hezbollah said warplanes and artillery had targeted insurgents in al-Waer after rebel sniper fire hit areas neighbouring the district.
In a little over a minute, they manage lift her free and carry her, alive, back into the house.
Rescuers can be heard repeatedly proclaiming "God is great, God is great" as young Aya was carried to safety.
Despite miraculously surviving the ordeal, the extent of her injuries are unclear.
Activists have reported air strikes in two other neighbourhoods, Qabun and Barzeh, over the weekend.
In Syria, the government forces fired rockets at a rebel-held area on Damascus's outskirts.
Regime forces shelled the Qaboun neighborhood in eastern Damascus for the second day, killing four people, the sources told Anadolu Agency. The Observatory estimates several thousand rebels remain there.
Ankara and Moscow have teamed up to host two rounds of Syria peace talks in Kazakhstan in a bid to bolster a faltering truce they brokered in December.
Since then, more than a quarter of a million people have been killed and more than 10 million displaced across the war-torn country, according to the United Nations, but the Syrian Center for Policy Research puts the death toll at more than 470,000.