The claim of responsibility came unusually swiftly for the group, which has been losing territory in Iraq and Syria.
Police have advised the public to steer clear of the area.
One police officer is dead and another hurt following a shooting in the French capital, police sources say.
The bustling shopping street in the heart of the city was blocked by armed officers and nearby metro stations closed. "We are on absolute vigilance, above all because of the elections".
Dozens of vehicles from the emergency services were sent to the area, where a helicopter was also deployed.
Earlier this week, two men were arrested in Marseille whom police said had been planning an attack ahead of the election.
"It looks like another terrorist attack", he said.
UPDATE (4:09 p.m.): President Trump offered his condolences for the people of France during a joint press conference with Italian Prime Minister Paolo Gentiloni.
He calls the attack a "terrible thing" and says "it never ends".
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The incident comes three days before the first round of one of the most tightly contested presidential elections in the country, and at a time when France is still under emergency rule following a wave of terrorist attacks in the past two years.
Eyewitnesses told SBS Europe correspondent Brett Mason that a second attacker was at the scene.
An officer has been killed and another hurt in a shooting on the Champs Elysées in Paris Thursday night, according to French media.
The Paris prosecutor's office said counterterrorism investigators are involved in the probe of Thursday's attack.
"I think this election is sufficiently unstable that it could still move things", he said.
France has been under a state of emergency since November 2015, when gunmen killed 130 people in attacks on a concert hall and other sites in Paris.
Up until now, polls showed voters more concerned about unemployment and their spending power than terrorism or security, though analysts warned this would change in the event of further bloodshed. "Marine Le Pen is notably one to watch".
"We are suffering the consequences of a laxity that has continued for years", she said shortly before the shooting, promising to take a hard line against extremists and anyone suspected of being an Islamist.
TV footage showed the Arc de Triomphe monument and top half of the Champs Elysees packed with police vans, lights flashing and heavily armed police shutting the area down after what was described by one journalist as a major exchange of fire near a Marks and Spencers store. It identified the attacker as Abu Yussuf al-Baljiki, suggesting he's from Belgium - details not confirmed by the French authorities.