The suspected terror attack is the latest in a string of assaults that have hit France since January 2015, claiming more than 230 lives.
The servicemen were hit by a BMW which drove down a quiet street in the upmarket western Paris suburb of Levallois-Perret at around 8:00 am.
It accelerated as it neared the troops, rammed into them and then sped away.
"I heard a huge crash which I thought was the sound of scaffolding being put up," Thierry Chappe, a resident in a building opposite the crime scene, told AFP.
Police later gave chase to the vehicle on a motorway north of Paris, and shot and wounded the suspect, a man aged in his late 30s who was also arrested, sources involved in the manhunt said, speaking on condition of anonymity.
Interior Minister Gerard Collomb described the act as "deliberate" and carried out by a "man on his own".
He spoke after visiting three of the injured soldiers along with Defence Minister Florence Parly. All six were taken to hospital, but none has life-threatening injuries, they said.
They were part of the 7,000-strong anti-terrorism Sentinelle force set up in January 2015 which sees armed, uniformed soldiers patrol the streets and guard high-risk areas such as tourist sites and religious buildings.
Collomb said the forces had been attacked on six different occasions since 2015. The country has been under a state of emergency since major attacks in Paris in November 2015.
The Paris prosecutors' office said its anti-terrorism unit has launched a probe into "attempted killings... in relation to a terrorist undertaking".
The Islamic State group (IS) has repeatedly targeted France because of its participation in the US-led international coalition fighting the jihadist group, with French jets carrying out air strikes in Syria.
"Security forces are actively seeking the perpetrator who is on the run. The ongoing probe will determine his motives and the circumstances in which he acted," Parly said in a statement earlier.
The incident came just four days after Sentinelle soldiers intervened to control an 18-year-old with a history of psychological problems at the Eiffel Tower where he brandished a knife and shouted "Allahu Akbar" (God is Greatest).
He told investigators he wanted to kill a soldier, sources close to the case told AFP.
In February, a man armed with a machete attacked four soldiers on patrol at Paris's Louvre Museum, while in April another extremist shot and killed a policeman on the Champs-Elysees, the French capital's most famous boulevard.
In June, a 40-year-old Algerian doctorate student who had pledged allegiance to ISIS attacked a policeman with a hammer outside Notre Dame cathedral.
The wave of attacks in France has had a serious impact on tourism in the world's top tourist destination, but the industry has begun to recover as incidents have become more widespread and generally less deadly.
The attack took place as the new centrist government was holding its last cabinet meeting before the summer holidays.
President Emmanuel Macron publicly clashed with the head of the French armed forces last month over a proposed cut to the military budget this year.
Macron slapped down General Pierre de Villiers, telling him "I'm your boss", after he complained about the impact of the cuts at a time when the army was in action in the Middle East and West Africa as well as at home. De Villiers resigned a few days later.