Opposition activists have accused the regime of orchestrating the explosions.
The al-Nusra Front emerged in January and has said it was behind previous attacks, including one in March on a police HQ and airforce Intelligence.
The video says the bombings were in response to attacks on civilian areas by forces loyal to President Bashar al-Assad.
"We fulfilled our promise to respond with strikes and explosions," a distorted voice says in the video, according to the Associated Press.
AnalysisThe group's statements echo those of jihadist groups, and the latest bomb attack was certainly similar to some in Iraq which have been blamed on al-Qaeda. But little else is known about al-Nusra. Who leads it, what its ideology is, and where it originated are just guesswork at the moment.
If this is the start of an al-Qaeda style bombing campaign in Syria, it will complicate an already intractable conflict. It would harden attitudes on both sides, and heighten sectarian suspicion.
The Syrian government has frequently accused the main opposition groups of links to al-Qaeda. So far there is no clear evidence of this. But the fact that al-Nusra says it shares their goal of overthrowing President Assad may make some countries which support the opposition feel uneasy about the possibility of al-Qaeda infiltration.
BBC security correspondent Frank Gardner says al-Nusra refers to its fighters as "mujahideen of Sham [Syria] in the arena of jihad" and there are suspicions it may have links to al-Qaeda.
The tactics used in the Damascus attacks are similar to attacks by al-Qaeda in Iraq in recent years, the BBC's Jonathan Head in Istanbul reports.
Meanwhile, the funerals of some of those killed in Thursday's blasts have been taking place in the city.
Violence has been continuing across the country despite a ceasefire being monitored by a team of UN observers.
The UN estimates at least 9,000 people have died since pro-democracy protests began in March 2011.