The bomber opened fire in the store before detonating his explosives, said police and witnesses.
Afghans and foreigners, including two women and a child, were among the dead, say reports. The Taliban told the BBC they had carried out the attack.
The blast left the Finest store, not far from the British embassy, ablaze.
The BBC’s Quentin Sommerville, in Kabul, says the supermarket is located next to a busy roundabout and a police checkpoint, and like many stores in the Afghan capital it has armed guards.
A police officer, Nazamuddin, told the BBC: “I was standing here when I suddenly heard a bang. After a few moments, I heard another bang. I didn’t go inside to find out what’s happened.”
The explosion in the heavily guarded Wazir Akbar Khan area of the city, an area frequented by foreigners and affluent Afghans, scattered debris across the road.
A witness told the BBC: “I was napping at my shop when I heard gun shots. Then we heard a loud explosion. Everyone was running around on the main road.”
An officer with the National Directorate of Security, Afghanistan’s spy agency, told the BBC that the guards had exchanged fire with the attacker for several minutes.
Mary Hayden, a consultant who was in the shop, told the Associated Press news agency: “To my left, I heard a gunshot. A bomb went off. Everyone was running to the back of the building.”
A Reuters news agency cameraman saw three dead bodies at the scene, including two women.
“We claim responsibility for the attack, and it was carried out at a time when foreigners were shopping, including the head of a security company,” Taliban spokesman Zabiullah Mujahid was quoted as telling Reuters.
A “ring of steel” erected around the capital before last September’s parliamentary elections has failed to prevent militant attacks.
Earlier this month in Kabul, at least four people died when a suicide attacker rammed his motorcycle into a bus carrying security forces.
Violence in Afghanistan is at its worst level since the overthrow of the Taliban government in late 2001.
US forces are due to begin withdrawing from the country in the middle of this year. – Bbcnews