The two suicide attacks took place at about midday local time (07:30 GMT) on Wednesday, targeting a police station and intelligence agency offices.
Taliban spokesman Zabihullah Mujahid claimed responsibility for the attacks.
It is the latest in a string of attacks to challenge the Afghan authorities after the resurgent militant group started its spring offensive early.
Condemning the bombings, President Ashraf Ghani said: "After the killing of [prominent commander] Mullah Salam and the Taliban's defeat on many other fronts, the terrorists are launching such attacks to raise the moral of their fighters."
Salam was killed in a US air strike on Sunday.
The first of Wednesday's attacks began when a suicide car bomber detonated his explosives outside a police station - which is next door to a military training facility - in the west of the city. This was followed by a five-hour gun battle between officers and another attacker.
Most of the fatalities reportedly occurred in this attack.
Soon afterwards a suicide bomber blew himself up outside Afghanistan's intelligence agency, in eastern Kabul.
Officials originally said just three people had died in the attack, but revised the number up later in the day.
The attacks came a day after the Taliban killed 12 policemen in an "insider attack" in the southern Afghan province of Helmand.

Two people were left with "life-changing" injuries when a corrosive substance was thrown on to them through their car windows.
Cousins Resham Khan and Jameel Muhktar had been celebrating Ms Khan's 21st birthday before the attack.
John Tomlin, 24, is described as white, 6ft tall, of stocky build and with short fair hair.
The new images show Mr Tomlin, who has distinctive hand and face tattoos, entering a shop in Romford on 3 July.
Mr Tomlin is also known to frequent the Canning Town area.
Ms Khan, a student at Manchester Metropolitan University, and Mr Muhktar suffered severe burns to the face and body in the attack on 21 June.
"Within seconds, my cousin started screaming. Her eyes were blistered, her face started melting. My face started melting", Mr Muhktar said.
According to police, they had stopped at traffic lights when the man approached them and threw the toxic substance at Ms Khan through the window.
The attacker then circled around the car and threw more of the acid at Mr Muhktar, before eventually fleeing the scene.
Mr Muhktar, 37, had to be temporarily placed in an induced coma to treat his injuries.
"I'm in a really bad way. I'm going to be scarred for life. I can't walk properly, I can't hear properly, I can't sleep", he said.
Police said the incident is being treated as a hate crime.
Almost simultaneous attacks in Kabul have left at least 16 people dead and 44 injured, the health ministry says.
[NEXT_CONCEPT]
Police have released new images of a man they want to talk to in connection with an acid attack in east London.