The assault was the latest in a wave of relentless militant attacks in Kabul this month by the Taliban and rival Islamic State militants that have killed scores and left hundreds wounded.
The attack started around 4 am, witnesses said, and fighting continued long after daybreak.
A suicide bomber first struck the military unit responsible for providing security for the academy, followed by a gunbattle with the troops, said Dawlat Waziri, spokesman for the Afghan defense ministry.
At least five insurgents were involved in the morning assault, according to Waziri. Two of the attackers were killed in the gunbattle, two detonated their suicide vests and one was arrested by the troops, he said.
All roads leading to the military academy were blocked by police, which only allowed ambulances access to the site to transfer the wounded to hospitals.
After the gunbattle ended, the security forces resumed control of the area. They also confiscated one suicide vest, an AK-47 and some ammunition, Waziri said.
Afzal Aman, commander of the city's military garrison, confirmed the attack in the area of the Marshal Fahim academy.
Hashmat Faqeri, a resident near the site, told AP he heard sounds of explosions and a gunbattle.
Attacks have increased in recent months in Kabul and elsewhere across Afghanistan, including massive bombings staged by militants determined to inflict maximum casualties, instill terror in the population and undermine confidence in Afghan President Ashraf Ghani's government and the country's security forces.
On Saturday, a Taliban attacker drove an ambulance filled with explosives into the heart of the city, killing at least 103 people and wounding as many as 235.
The Taliban claimed the ambulance attack, as well as an attack a week earlier in which militants stormed a hilltop hotel in Kabul, the Intercontinental, killing 22 people, including 14 foreigners, and setting off a 13-hour battle with security forces.
Masoom Stanekzai, the head of Afghanistan's intelligence service, said five suspects have been arrested for their involvement in the hotel attack. A sixth suspect had fled the country, he said.
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