The "hacktivist" collective vowed vengeance for attacks in Paris on Friday, claimed by IS, that left 129 dead and hundreds injured, some of them critically.
"Anonymous everywhere in the world is going to hunt you down," a hooded figure in black, wearing the group's signature Guy Fawkes mask said in French.
It was not possible to verify its authenticity, but the statement and video had the hallmarks of the network, known for mounting cyberattacks against government and corporate websites.
"We are going to launch the biggest operation every mounted against you -- get ready for a multitude of cyberattacks. War has been declared."
More From This Section
The video, posted the day after the attacks, had more than 1.3 million views by today afternoon.
In an apparent riposte, a message posted on the Twitter address of the messaging service Telegram calls on Islamic State affiliates to secure their Internet communications.
"The #Anonymous hackers threatened... That they will carry out a major hack operation on the Islamic state (idiots)," the message said.
Charlie Winter, a researcher in transnational jihadism at UK-based think tank Quilliam, affirmed the message's authenticity in a tweet.
"@GroupAnon, IS didn't like your declaration of war," he wrote, referring to one of Anonymous' Twitter addresses.
"Here's what they're saying on @telegram. Use it against them."
French cybersecurity expert Olivier Laurelli warned the Anonymous action against the jihadist organisation could interfere with police efforts to identify and track its members.
"It's counterproductive," he told AFP. Actions that force Islamic State operatives "to close accounts just renders police investigators blind and dead for certain things."