Maj. Gen. Dawlat Waziri, spokesman of the Ministry, told Anadolu Agency that the suicide bomber walked towards the bus and detonated himself, leaving 15 cadets dead and four injured in the Qambar Square area of the city.
Suicide bombers struck two mosques in Afghanistan during Friday prayers, a Shiite mosque in Kabul and a Sunni mosque in western Ghor province, killing at least 63 people at the end of a particularly deadly week for the troubled nation.
"We commend the government and security forces of Afghanistan for their response to these attacks, and we offer our deepest condolences to the families and friends of those who were killed", the spokesperson said.
The suicide attack inside the Imam-e-Zaman Mosque in Kabul claimed at least 50 lives, with 50 more wounded, the statement said, adding that all of the victims were civilians.
Suicide bombers on Saturday killed 15 cadets when they attacked Marshal Fahim National Defense University in Kabul.
No group immediately took credit for the attack, but the Islamic State militant group has recently claimed responsibility for similar attacks.
Residents say attendance at local Shia mosques in Kabul on Friday has dropped hugely due to fears of being targets.
Around seven major attacks have been carried out in Afghanistan by Taliban or Islamic State group militants since Tuesday.
The Sunni Muslim group did not provide evidence for its claim but it has attacked Shia mosques before. Most of those attacks targeted Afghan police or military facilities. The Kabul attack marks the sixth on a Shiite mosque this year, including one in Herat province in August that killed 31.