Two other districts were seized by the militants in the past fortnight and several more have been threatened, according to The Long War Journal, which closely tracks the security situation in Afghanistan.
At least six people were wounded although it is feared casualties could be higher.
As police cordoned off part of the city's main commercial area, an official from a branch of Afghanistan International Bank reached by telephone said customers were sheltering in the bank premises as gunfire continued in the street outside.
The Kabul blasts have come two days after at least 13 people died following an explosion at a mosque in eastern Afghanistan. The two are believed to be among the five dead reported by the Public Health Ministry.
Afghan security personnel inspect the area of a suicide attack in the centre of Kabul Credit Rahmat Gul AP
The ISIS group later claimed responsibility for one of the attacks.
Kabul police chief Gen. Daud Amin says another two policemen were wounded in the attack.
The second attack took place in front of a police station in Shar-e-Naw neighbourhood in downtown Kabul where a "search and clearing operation" was still under way.
The attacks came as a strong 6.2-magnitude quake rocked Afghanistan, shaking buildings in Kabul where people were already on edge.
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Suicide bombers and gunmen launched apparent coordinated attacks on two Kabul police stations on Wednesday, with witnesses reporting on social media that both assaults are still under way.
Three blasts hit the Afghan capital of Kabul in rapid succession on Wednesday, police and interior ministry officials said.
"We also have reports of small arms fire".
Daesh claimed responsibility for the first attack on a police station in a heavily Shiite-populated neighbourhood in the city's west, which police spokesman Hashmatullah Estanakzai said had ended. However, the head of the Kabul ambulance service told the AP news agency that six people had been taken to hospitals with injuries.
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The Taliban and IS frequently target Afghanistan's Western-backed government and security forces.
The Taliban insurgency claimed its members carried out the attack and killed many officers and personnel of Afghan security and intelligence agencies. He said another explosion occurred near a different police station.
The Taliban have confirmed in a statement they control the district and say they also killed 10 government forces.
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