Ten of the victims were killed by air strikes and another four died in rocket fire, the Britain-based Syrian Observatory for Human Rights said.
It said another 40 people were wounded and the death toll was likely to rise.
At a medical centre near the town of Hazza, an AFP photographer saw wounded children and a child's body wrapped in plastic.
Saturday's government raids come after an attack on Tuesday by the Islamist rebel group Ahrar al-Sham on a military base near the town of Harasta.
On Friday, regime bombardments killed at least 19 people, including six children, mostly in the city of Douma.
The deaths came amid an escalation in tit-for-tat attacks between regime forces and rebels holding the enclave on the capital's eastern outskirts.
Retaliatory shelling of Damascus on Thursday and Friday by rebels killed nine people.
The official SANA news agency said on Saturday rebel shelling of the city killed one person and wounded 20.
Eastern Ghouta is supposed to be part of a "de-escalation zone" under a deal between Russia, Iran and Turkey aimed at reducing the level of violence.
President Bashar al-Assad's forces have besieged Eastern Ghouta since 2013, and humanitarian conditions in the area, where some 400,000 people live, are dire.
More than 330,000 people are estimated to have been killed in the Syrian war, which began in 2011 as the regime brutally crushed anti-government protests. Millions have been displaced.