The group calling itself the Cyber Army of the Khilafah (Caliphate) replaced the front page of the war monitor's site with a photoshopped image of the Observatory's director and text threatening him.
The Observatory is a Britain-based monitor of the war in Syria, and one of the few groups that has reported on violations by all sides in the conflict and documented its ongoing death toll.
A screenshot of the hacked website provided by the Observatory showed an image of director Rami Abdel Rahman's face photoshopped onto the body of an IS hostage dressed in orange and seemingly about to be beheaded by a knife-wielding jihadist.
The text claims that the "Cyber Army of the Khilafah broke into the computer systems of SOHR... seized control of its website, destroying it, and wiping out the data."
Abdel Rahman confirmed the hackers had destroyed data on the Observatory's servers, but said "we have a copy of all the information that has been published and until the site is back up, we will publish on our Facebook and Twitter accounts."
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He said the Observatory had previously received similar threats from the Syrian government and Al-Qaeda affiliate Al-Nusra Front.
"But we will continue to do our work and document what is happening in Syria," he told AFP.
There was no way to verify that the hackers were linked to IS.
A similar hacking operation against a French television station this year may have originated in Russia.
The April attack against France's TV5Monde was claimed by a group calling itself the "CyberCaliphate," but in June a judicial source said an investigation into the attack was focusing on a group of Russians.
Hackers claiming to be affiliated with or supporters of the Islamic State group have previously hacked the Twitter account of US Central Command and other international websites.