The incident took place in the vicinity of the first police district of the city at around 6:20 pm local time.
A spokesman for the Kabul police Basir Mujahid confirmed that unknown armed individuals hurled at least three hand grenades on a security post close to Spinzar hotel.
He said at least one civilian was killed in the attack and three security personnel were wounded.
No individual or group has so far claimed responsibility behind the incident.
This comes as at least one person was killed in a magnetic bomb explosion earlier in the same day.
According to the ministry of interior officials, the incident took place in the vicinity of the 8th police district of the city.
Najib Danish, spokesman for the ministry of interior, said a magnetic bomb planted in a Toyota Fielder type vehicle went off at around 7 am on Tuesday, leaving the driver of the vehicle dead.
The anti-government armed militant groups have stepped up their efforts to carry out attacks in the key provinces and cities of the country, including capital Kabul.
The Afghan intelligence, national directorate of security, said late on Monday that two suicide bombers were arrested before they manage to carry out a car bomb attack in the city.
Meanwhile, the security forces confiscated a mini-truck packed with explosives from the vicinity of the 5th police district of the city a day earlier.