Ali Shah Paktiawal, a police official, said the bomber was on foot and detonated his explosives next to the walls of the Babur Gardens.
He said the people who suffered injuries were riding in a minibus.
There were no police, soldiers or foreigners in the area at the time of the attack, Paktiawal said.
Bashary said the attacker was apparently trying to drive into the city centre to carry out the bombing.
When he saw the checkpost, less than 3km outside the heart of Kabul, he got out of his car and blew himself up, Bashary said.
"The bomber was shattered into small pieces," he said.
There was no immediate claim of responsibility for Tuesday's blast.
The last suicide attack in Kabul was on July 7, which severely damaged the Indian embassy and killed more than 50 people, including two senior diplomats.
Elsewhere in Afghanistan, US-led forces said they called in air strikes after a firefight on Tuesday that they said resulted in the killing and wounding of 25 Taliban fighters.
The reported battle - the latest in a series of clashes between US forces and the Taliban - took place in Farah, a western province.
Two Afghan police officers were wounded in the fighting, while there were no reports by US forces of casualties on their side.
According to the Associated Press, battles with the Taliban have resulted in a toll of 2,500 deaths this year alone.