The attack comes a week after a French policeman was shot and killed and two others wounded on the Champs-Elysees avenue in Paris, just days ahead of the first round of the presidential election.
More than 230 people have been killed in a string of jihadist attacks on the French mainland since January 2015 and the country has been under a state of emergency for nearly a year and a half.
"The man refused to be arrested and fired a rifle at police," a government official said, adding that the suspect - in his 20s, who is believed to be a recent convert to Islam - was now in custody.
The man lived with his mother, also in custody, in an apartment in Saint-Benoit, in the island's east.
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Authorities seized several weapons and other material that could be used to make Molotov cocktails, France's interior and overseas ministers said in a joint statement.
The suspect "is a very discreet man who didn't seem to have a lot of friends and was never a problem," a neighbour said.
"We did notice that he had been sporting a beard lately but we didn't pay much attention to it," added a young man.
Authorities in Reunion estimate there are around 100 radicalised Islamists on the island.
The string of terror attacks in France began in January 2015 with a massacre at the offices of the Charlie Hebdo satirical magazine.
The following November, IS gunmen and suicide bombers killed 130 people in Paris, and a Tunisian man rammed a truck through crowds in Nice last July, killing 86 people.
France remains on high alert as it prepares to choose between centrist Emmanuel Macron and far-right leader Marine Le Pen in the second round of the presidential election on May 7.