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.
Almost simultaneous attacks in Kabul have left at least 16 people dead and 44 injured, the health ministry says.