A second bus was hit by a suicide bomber elsewhere in the city, injuring two soldiers and two civilians.
The Taliban said they had carried out the attacks.
The blasts come a day after Afghan and US officials signed a deal letting US troops stay in Afghanistan after the end of 2014.
The agreement was authorised by new Afghan president Ashraf Ghani, who was sworn in on Monday.
On that same day, two bombs killed at least 15 people in Kabul and Paktia provinces.
Mr Ghani's predecessor, Hamid Karzai, had refused to sign the deal because of a disagreement with the US.
The attack on the army bus took place in western Kabul, and largely destroyed the bus.
"I was walking down the road when a huge explosion occurred," shopkeeper Sanaullah Khan told Reuters.
"Right after the blast, I saw dust and smoke rising from the area and then I escaped."
The second attack took place in northeast Kabul.
The Taliban claimed responsibility for both attacks on Twitter, saying they were a continuation of an offensive which started in the summer.
Under a US-brokered unity deal Mr Ghani shares power with runner-up Abdullah Abdullah who becomes chief executive. The Taliban has called the deal a "US-orchestrated sham".
At least seven Afghan soldiers have died and many were injured after a suicide bomber targeted a bus carrying troops in the Afghan capital Kabul.