The raids come a week after gunman Mohammed Merah, who killed seven people, was shot dead after a long siege in the southwestern city of Toulouse.
The arrests took place in Toulouse, Marseille, Nantes, Lyon and the Ile de France region, around Paris, the Interior Ministry said.
Authorities have not said that any of those arrested were directly linked to Merah.
Sarkozy told Europe 1 that the decision to act had been taken by the interior minister and foreign minister "to deny the entry of certain people to France" who did not share the country's values.
"It's not just linked to Toulouse. It's all over the country. It's in connection with a form of radical Islam, and it's in agreement with the law," he said.
"What you have to understand is that the traumatic events in Montauban and Toulouse were profound in our country. I don't want to compare horrors but it's a bit like the form of trauma visible in the United States and New York after 9/11. We have to be able to draw some conclusions."
Interior Minister Claude Gueant said that several firearms, including five rifles, four automatic weapons and three Kalashnikovs, had been found in the searches, as well as a bulletproof vest.
Speaking to French media, he said the raids targeted people who have made the claim online that they are "mujahedeen," or Islamist fighters, and support "an extremely radical ideology."
The authorities' decision to swoop was in part based on the suspects' claims that they had received paramilitary training, Gueant said.
Sarkozy suggested that more raids will follow, saying, "There will be other operations that will continue and that will allow us to expel from our national territory a certain number of people who have no reason to be here."
Sarkozy said he was obliged to act to ensure the nation's safety. "It's our duty to guarantee the security of the French people. We have no choice. It's absolutely indispensable."
The Interior Ministry media office said "the police had plans to carry out 19 arrests, and therefore 19 arrests were made in connection with the group Forsane Alizza."
Forsane Alizza is a pro-al Qaeda group in France with a cluster of followers in Toulouse. Merah appeared to have developed connections with the group, according to French media reports.
The group was outlawed in January for encouraging French citizens to travel to Afghanistan to fight jihad.
Police have been investigating whether Merah acted alone in planning his attacks.
He is blamed for the killings of three French paratroopers, a rabbi and three Jewish children ages 4, 5 and 7. Two other people were seriously wounded in the shootings.
Merah told police he had attended an al Qaeda training camp while visiting Afghanistan and Pakistan, according to Paris prosecutor Francois Molins.
But his uncle, Jamal Azizi, denied statements by French authorities that Merah was an al Qaeda sympathizer and that he had traveled to Afghanistan or Pakistan to train to use arms.
Merah was buried Thursday at a cemetery outside Toulouse. Algeria, where his family is originally from, had refused to accept his body, Merah's father said, citing French authorities.