The suspected terror attack follows a string of assaults that have hit France since January 2015, claiming more than 230 lives.
The suspect, a 36-year-old Algerian man, has been hospitalised after police shot him five times.
The servicemen were hit by a BMW which drove down a quiet street in the upmarket western Paris suburb of Levallois-Perret at around 8am (1pm Thai time).
It accelerated as it neared the troops, rammed into them and then sped away. Three were shocked and lightly hurt, while the other three sustained more serious injuries which are not life-threatening, officials said.
“I heard a huge crash which I thought was the sound of scaffolding being put up,” said Thierry Chappe, an AFP employee who lives in a building opposite the crime scene.
Some 300 police later tracked the rented vehicle to a motorway near the northern port of Calais.
After a chase, officers opened fire, wounding the unarmed driver who was arrested then taken to hospital, sources involved in the manhunt said.
The suspect lived in the Yvelines suburb of Paris and had no previous convictions.
Pictures showed the black BMW with a crumpled front end and smashed windscreen on the A16 motorway.
An AFP journalist saw masked police carrying out a search in Bezons in the northwestern Paris suburbs.
The servicemen targeted in the assault were part of the 7,000-strong anti-terrorism Sentinelle force set up in January 2015 after jihadist gunmen attacked the offices of satirical weekly Charlie Hebdo, killing 12.
The armed, uniformed soldiers of the force patrol the streets and guard high-risk areas such as tourist sites and religious buildings.
President Emmanuel Macron congratulated security forces on Twitter for apprehending the suspect, adding: “Vigilance remains a duty at all times.”
Interior Minister Gerard Collomb said Sentinelle forces had been attacked on six different occasions since 2015. France has been under a state of emergency since the Islamic State attacks in Paris in November 2015, which left 130 people dead.
Some experts believe patrols should be withdrawn from the streets where they are an obvious target for extremists, but the government is likely to hesitate to do so in case of another major attack.
“The opposition would jump on the argument that ‘you’ve dismantled Sentinelle, there’s been an attack and you are to blame’,” said Alain Rodier, a specialist from private national security organisation Cf2R.
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 jihadists, with French jets carrying out air strikes in Syria.