The first bombing in Kabul took place around 8.10 a.m. when a suicide attacker riding a motorcycle targeted a bus carrying employees of the Mines and Petroleum Ministry who were on the way to their office, said Kabul police spokesperson Firdaws Faramarz.
The second explosion was a magnetic bomb that targeted the first responders arriving on the scene, said Interior Ministry spokesman Nasrat Rahimi, amid fears that the death toll may increase further.
The third attack took place when one more suicide bomber blew up his explosives-laden vehicle in a different Kabul neighbourhood, a mostly-residential area that also has several factories.
At least 45 people were injured in the bombings, Efe news reported.
The Taliban claimed responsibility for the third blast and said it had targeted a convoy of foreign forces, but denied involvement in the first two bombings. Rahimi, however, said the militant group was believed to be behind all three blasts.
Condemning the "barbaric attacks", Afghan President Ashraf Ghani said the Taliban was on one side holding peace talks and on the other continuing "crimes against humanity".
The spike in attacks, the President said, won't give the Taliban greater leverage in peace talks with the US.
Taliban spokesman Zabihullah Mujahid in a statement said "nine high-ranking foreign officers" were killed in the third bombing, a common exaggeration for the militant group about casualties inflicted on security forces.
The NATO mission in Afghanistan has, however, denied that its troops were targeted in Kabul, according to an international mission spokesperson.
The triple attack in Kabul took place in the middle of the visit to Kabul by the US Joint Chiefs of Staff Chairman Marine General Joseph Dunford for meeting top American and NATO commanders in the city.
US peace envoy Zalmay Khalilzad, leading the American delegation for talks with Taliban militants to end the 18-year war, was also in the city to meet Afghan officials before he begins the latest round of peace dialogue with militant leaders in Qatar.
Khalilzad met the Afghan President 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 )