In a statement circulated on its social media accounts, the group said three of its fighters armed with guns, grenades and explosives were involved in the attack in the southern neighbourhood of Midan yesterday.
The Syrian interior ministry said yesterday that the attack involved two suicide bombers, one of whom managed to penetrate the police station and reach the first floor before his explosives detonated.
But a monitor reported a third explosion involved a car bomb outside the police station, and the IS claim also referred to the third attacker blowing himself up separately from the other two.
Damascus has been largely insulated from the worst of the violence during the country's brutal six-year civil war, but several bomb attacks have shaken the city.
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The Midan police station has itself been targeted before.
In December 2016, three police officers were wounded when a seven-year-old girl walked into the police station wearing an explosive belt that was remotely detonated.
The regime is currently waging several offensives against IS, including in the Badiya desert region and in the eastern province of Deir Ezzor.
More than 330,000 people have been killed in Syria since the conflict began in March 2011 with anti-government protests.