Neighbours calling police claimed they heard the man chanting Islamist slogans.
The shooting happened about 9 pm local time when a car stopped at 102 Champs-Elysees in front of a police van, Interior Ministry spokesman Pierre Henry Brandet said. A man emerged from the car and opened fire on the van with an "automatic weapon," killing one officer instantly, he said.
European soccer's governing body warned the English and Russian football associations on Sunday that if fan violence continued their teams could be disqualified from the Euro 2016 tournament.
European soccer's governing body, UEFA, launched disciplinary proceedings against the Russian Football Union on Sunday after violence marred the end of Russia's Euro 2016 match against England in Marseille on Saturday evening.
At least 10 children are among the dead in the Riviera city following a "cowardly and barbaric" atrocity believed to have been carried out by 31-year-old Mohamed Bouhlel.