The first blast occurred at 07:50 a.m. local time after a bomb struck a running bus in Police District 5, spokesman of Ministry of Interior Affairs told Xinhua.
"The targeted bus was carrying Burhanuddin Rabbani Education University lecturers and staff members. One person was killed and 10 injured in the incident. The injured were shifted to hospitals by police vehicles and ambulances," spokesman Nasrat Rahimi said.
The explosion was caused by a sticky bomb attached to the vehicle, he said.
"We heard a huge blast in the eastern edge of Khoshal Khan neighborhood. The blast was followed by two other explosions. The whole place has now been sealed off," an eyewitness said.
The second and third blasts came after police, rescue team, people and reporters arrived at the site. One local journalist was among the injured, he said.
No group has claimed the incident yet.
Over the past months, Afghanistan, especially Kabul, has witnessed waves of terror attacks by the Taliban insurgents and Daesh (ISIL or ISIS).