The third vehicle, however, was able to make it into the city center close to Tahrir Square where the bomb went off.
The suicide bomber was driving a vehicle as authorities chased him at Al-Tahrir square, but he escaped and detonated his explosives.
One of the suicide bombers blew himself up in Tahrir square in central Damascus after being encircled by the authorities.
The minister for local governments and environment Hussein Makhlouf said from the site of the attack that the suicide bombers wanted to target more populated areas, especially that people are back to work following the Eid holidays.
Syrian State TV showed burnt out cars and buses and damaged buildings along the road.
No group immediately claimed responsibility for the attacks, but the jihadis of Islamic State and militants linked to al-Qaeda have claimed recent attacks inside the city.
According to State TV, the attackers planned to bomb crowded areas in the capital on the first working day after the Muslim holy month of Ramadan.
The Britain-based Syrian Observatory for Human Rights, an opposition-run monitoring group, also reported the three explosions.
It was a rare attack on Damascus - the seat of President Bashar Assad's government - despite the Syrian War having raged on in the country since 2011.
Conflict continues in the east of city between rebels and security forces.
Twin blasts left 74 dead in the capital's Old City on March 11 and were claimed by former Al-Qaeda affiliate Fateh al-Sham Front, known before as the Al-Nusra Front.