The first explosion took place around 8.10 a.m. when a suicide bomber riding a motorcycle targeted a bus carrying employees of the Mines and Petroleum Ministry, Kabul police spokesperson Firdaws Faramarz told Efe news.
The second bomb was detonated near the bus attack site, the Interior Ministry said in a statement.
At least seven people were killed and 21 wounded in the attacks, Public Health Ministry spokesperson Wahidullah Mayar said in a statement.
Amid fears that the death toll due to the bombings may increase, the Interior Ministry said that the victims of the two attacks included women and children.
The third attack took place when an explosives-laden vehicle was detonated in another part of the city, a mostly-residential area that also has several industrial factories.
But the target of the third bombing was still not clear. It was also not known immediately if there were any casualties in that attack, according to the Interior Ministry.
No group claimed the attacks but the government suspected that the bombings were carried out by the Taliban.
"Taliban violence against civil continues across the country. The Ministry of Interior strongly condemns today's despicable and senseless attack (...) The deliberate targeting of civil is a war crime," the official statement said.
The triple bombing in Kabul took place less than a week after nine people were killed and 33 injured in an attack on July 19, when a suicide bomber detonated his explosive-laden car near Defence Ministry facilities while being chased by the police.
The attacks also occur in the midst of the ongoing visit of the US' envoy for Afghan reconciliation, Zalmay Khalilzad, who met with Afghan President Ashraf Ghani and Chief Executive Abdullah Abdullah on Wednesday to help form a team to negotiate with Taliban leaders.
After his Kabul visit, the envoy is set to leave for Qatar's capital, Doha, where the eighth round of peace talks with Taliban leaders to negotiate the withdrawal of international troops - the militant group's main demand - is scheduled to take place.
Both sides are seeking a way out of nearly two decades of Afghan war, although the Taliban have so far refused to directly negotiate with the Afghan government.
( With inputs from IANS )