The clashes -- which broke out Sunday in the Syrian capital -- were reported by opposition groups, the Syrian Observatory for Human Rights and Syrian state media.
Damascus has not seen such attacks in months if not years, said SOHR Director Rami Abdel Rahman.
Since the start of the conflict, the Syrian regime and allied militias have held a tight grip on the capital. The opposition has only a few pockets of control around Damascus, including a part of the district of Jobar. The rebel fighters began the attack from that area of the city.
The assault was led by Islamist groups including Ahrar al-Sham and the recently formed umbrella jihadist group Hayat Tahrir al-Sham. That group includes another -- Jabhat Fatah al-Sham.
Syrian state media reported that Syrian forces had repelled an attack by jihadist fighters, who also used tunnels to attack government positions.
ISANA, the Syrian state-run news agency quoted a military source Monday saying the regime has recaptured all positions in the industrial area that armed groups infiltrated on Sunday.
However, Faylaq al-Rahman, one of the rebel groups taking part in the Damascus fighting, issued a statement Monday saying the attack was ongoing in the capital.
There was no word on deaths.
The clashes come at a time when the Syrian regime has been weakened in the capital -- located in the south of the country -- due to its focus on fighting Isis in Raqqa
, in the north, said Andrew Tabler, a senior fellow in the Arab politics program at the Washington Institute for Near East Policy.
"The regime only has about 18,000 in deployable manpower and they're being pushed by Russia and the Iranians to go and fight for Raqqa. This has left them vulnerable elsewhere," he said.
"They're spread too thin and this has allowed rebels to advance."
Lina Khatib, the head of the Middle East and North Africa program at Chatham House said this is the first attack in Damascus of its kind by rebel groups that, until now, had been operating mainly in northern Syria.
She sees it as a signal that one of them, Hayat Tahrir al-Sham, is raising its profile.
"This attack in Damascus will show other rebel groups it is capable of delivering results," she said.
The assault comes after a series of suicide bombings that have hit the capital in March.
On Wednesday, suicide bombings struck a courthouse
and restaurant in Damascus, killing more than two dozen people and injuring others, Syrian state news said.
What happens next will depend how Russia and Iran respond, according to Haid Haid, a Syrian writer and researcher who is an associate fellow at Chatham House.
"The most likely scenario is that the fight will continue for some time until there's some kind of agreement what to do there -- a military solution among the regime and its allies or a political one to de-escalate."