Police sources said the officer sustained only minor neck injuries in the assault, which comes with France on high alert after jihadists killed seven people in London on Saturday.
The suspect later claimed to be a "soldier of the caliphate" of the Islamic State group, according to a source close to the investigation.
The policeman's colleague opened fire on the man, hitting him in the chest in panicked scenes around the Gothic cathedral that is one of France's most visited tourist attractions.
About an hour after the attack he was taken to hospital and police declared the situation to be under control.
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Interior Minister Gerard Collomb said the man had shouted "this is for Syria" as he lunged at the officer.
He said the man had also been carrying "kitchen knives" and was found in possession of a card identifying him as an Algerian student.
Anti-terrorist prosecutors were put in charge of the investigation.
A witness told AFP he heard someone "shout very loudly".
Notre Dame, which is situated on the banks of the Seine river in the heart of Paris, draws 13 million visitors a year.
Over 1,000 people were inside the cathedral at the time of the attack.
Authorities in Paris asked the public to stay away from the area.
Pictures on social media showed people sitting in the pews with their hands in the air -- apparently at the request of police.
Andre Finot, the cathedral's head of communications, described the situation as calm. "People are talking to each other, praying and continuing their visit," he said.
The car was tracked to an all-female terrorist cell, allegedly acting on the orders of Syria-based Islamic State jihadists.