At least 14 people killed and dozens wounded by explosion at Cilvegozu border crossing in southern Hatay province.
Rebels have captured Syria’s biggest hydro-electric dam and battled army tank units near the centre of Damascus, activists say.
Monday’s claim came as the opposition renewed an offer to negotiate the departure of President Bashar al-Assad.
The rebel seizure of the Taqba dam, a prestige project on the Euphrates River in the eastern Raqqa province completed by Assad’s father in the 1970s, may have only limited impact on Syria’s already unreliable power supplies.
Activists said opposition fighters had overrun army positions around the dam a day earlier.
“The dam was protected by an artillery battery and many intelligence units. The rebels moved on them in a lightning offensive yesterday, overrunning their positions and capturing scores of personnel,” Abu Ziad Teif, an opposition activist in contact with rebels in the area, said.
Meanwhile, in Damascus, Assad’s forces brought up tanks to defend an area just east of the city centre on Monday, residents said.
Jobar, a district adjacent to the landmark Abbasid Square, has seen fighting in recent days. However, activists said Assad’s forces remained well dug in in the city centre.
“The main battle is taking place in Jobar,” an opposition activist in Damascus named Amer said. “The rebels appear to be advancing in the eastern sector.
“But the centre of Damascus is crisscrossed with concrete barriers and security is deployed everywhere; we cannot say that they [the rebels] have a real active presence in the centre.”
Against this backdrop, Mouaz al-Khatib, the head of the opposition Syrian National Coalition, said he received no response to his offer to discuss a handover of power with the Assad government, but that the invitation still stood, despite the passing of an initial deadline for a response on Sunday.
“The regime has not given a clear answer so far, clearly, frankly, that it accepts leaving to spare destruction and blood,” he said.
“No meetings have been arranged, and no formal contact with any party has happened so far.”
Pressed to say whether his offer was still open despite the deadline passing, Khatib said: “We are still waiting for the government response and then we are going to study that.”
Assad has given no sign of wanting to negotiate his own departure.
Syria’s state news agency quoted Assad as saying: “Syria will not give up on its principles however great the pressure and the conspiracies, which do not target Syria alone but all Arabs.”
Meanwhile, Riad Hijab, Assad’s former prime minister, met the Egyptian foreign minister in Cairo.
Quoted by Egyptian news agency MENA, Hijab, the most senior government defector from Damascus, said: “There is no solution to the Syrian crisis except by the departure of Bashar al-Assad.”
Hijab also said the coalition was seeking Syria’s national seat at the Arab League.
An Arab League source said the group would consult its other members on the request.
Syria’s membership in the pan-Arab body was suspended in November 2011.