The car tried to crash through the barriers at Place de la Concorde, and police opened fire as it sped away.
One of the two is a 16-year-old male who said he was at the wheel.
A third person, a young woman in hospital with bullet wounds, is also thought to have been in the car.
She was operated on overnight at the hospital in a suburb to the west of the French capital.
Her life is not in danger, a police source told journalists.
A bullet-ridden car was later found abandoned near to the incident. Two men and two women were reported to have been seen leaving the vehicle.
British cyclist Chris Froome won his second Tour de France later on Sunday.
Paris officials, investigating why a car was driven through the security barriers ahead of the Tour de France on Sunday morning, are holding two people who handed themselves in overnight.