The identity was not released. The attacker was also reportedly killed following the attack.
French far-right presidential candidate Marine Le Pen tweeted, "My emotions and solidarity for the police, once again targeted".
Conservative contender Francois Fillon said on France 2 television he was canceling his planned campaign stops on Friday.
If no candidate claims a majority in the first round - which is likely - there will be a run-off between the top two candidates on May 7.
The Islamic State group quickly claimed responsibility for the attack.
In a statement from its Amaq news agency, the group gave a pseudonym for the shooter, Abu Yusuf al-Beljiki, indicating he was Belgian.
A Reuters reporter saw a helicopter flying low over central Paris, apparently part of a follow-up police operation.
One policeman has been killed and a second wounded in a shooting in central Paris, police say.
One police officer and an attacker are dead after a shooting in Paris.
"We don't have a precise idea of the origin of the shots fired", he said.
The officials had no other details about the attacker. He was shot to death after opening fire on a police van.
The French prosecutors' office said the counter-terrorism office had opened an inquiry. Police sources had said earlier that the shooting could have been an attempt at an armed robbery.
BFMTV reported earlier that two policemen were seriously injured.
Paris police spokeswoman Johanna Primevert says unidentified gunman appeared to be alone when he allegedly opened fire on a police vehicle.
French President Francois Hollande scheduled an emergency meeting on Thursday night.
The shooting happened as the country is under a state of emergency from a series of extremist attacks.
The Champs-Elysees is a main road lined with restaurants, cafes, exclusive designer boutiques and tourist shops.
French television networks reported that the chief suspect was a 39-year-old French national known for previous violent crimes. "There were tens, maybe even hundreds of them".
Security is high after police said they arrested two men on Tuesday in what they described as a thwarted terror attack. "It never ends", the President said.
The Paris police department has advised people to avoid going to the area on its Twitter account.
Subway stations in the area were closed off on Thursday night while police secure the scene.
France is now under a state of emergency following a spate of Islamist terror attacks.
A massive police presence remained at the Champs Elysees long after the attacks. A man emerged from the auto and opened fire on the police van with an "automatic weapon", he said. The gunman tried to run, and wounded two other officers as he fled before he was killed.
Brandet says the police officers were "deliberately" targeted.
Paris resident Daoud Kal, 29, said he was walking in the area near a metro station when he heard four to five shots.
President Donald Trump offered condolences to the people of France during a joint news conference with the Italian prime minister.
The shooting comes just three days before the French election.