The Islamic State claimed responsibility for the attack.
A police spokesman said the attacker may have been as young as 12.
A defence ministry spokesman said that the blast happened outside their building.
The target of the attack was not immediately clear but as well as foreign embassies, the area contains a defense ministry department dealing with foreign relations and several private businesses, including a barber and a travel agency.
The explosion was reported by the TOLOnews television channel.
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Many injured people were carried from the scene of the blast, put into ambulances and police pickup trucks and driven away.
A public health official told TOLO News that three dead bodies and 21 injured had been taken to city hospitals. The suicide attack took place near a gate of the academy, near Qambar square.
The last major assault in Kabul was on Oct 20 when a suicide bomber blew himself up inside a Shiite mosque during evening prayers, killing 56 and wounding 55.
Initial reports suggest that the suicide bomber detonated explosives inside Kabul Green Zone close to Australian embassy.