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.

Research suggests Oxford may be affected the most by public sector cuts
Oxford among worst GCSE results
Chairman of Oxfordshire Economic Partnership Frank Nigriello said small firms would not have jobs available.
It comes as research suggests Oxford has one of the highest number of employees in vulnerable sectors.
The BBC-commissioned study, carried out by Experian, ranks the resilience of council areas to economic shocks.
In the research, Oxford was rated the third highest in England for the number of employees working in vulnerable sectors.
Figures from the government estimate 48% of jobs in Oxford East are in the public sector.
Mr Nigriello said Oxfordshire had very few large companies, with many people employed in the county's pioneering health services, as well as research and development, including Oxford University.
He added: "If there are massive cuts in the public sector, then Oxfordshire's private sector won't have the resilience.
"Small businesses don't have the opportunities or availability of jobs."
A special BBC News season examining the approaching cuts to public sector spending
The Spending Review: Making It Clear
The Oxfordshire Economic Partnership (OEP) is a company formed by a number of business people in the county to help promote enterprise.
The research also suggests Oxford has few self employed people, but that the working-age population were highly skilled.
In South Oxfordshire, according to the Experian research, the situation was different, with the 11th highest proportion of those self employed in England.
Mr Nigriello, who works for Unipart in Cowley, suggested the private sector should be willing to assist the public sector in making changes to their organisation.
He said: "Private sector companies would be foolish not to help.
"While we may be the engine that creates the wealth, we recognise that we need the services to create the community we want to live in".
Oxfordshire County Council is currently looking to make more than Â£200m of cuts by 2015.
The figure may be adjusted after the government's spending review next month.
Meanwhile, the research also revealed Oxford is in the bottom five regions of the UK for GCSE results.
South Today will be debating the issues raised in the spending review tonight at 2235 BST.
BBC Oxford 95.2FM will be discussing the research from 0700 BST.

It had already been announced the 30-year-old would leave the Warriors when his contract expires this summer.
The South Africa-born forward becomes Sale's first new signing for next season.
"Josh is a world class ball-carrying forward," said Sharks director of rugby Steve Diamond.
"He has been performing well for both Glasgow and Scotland over the past five years and will complement the squad in the ball-carrying department."
Strauss joined Glasgow from Super Rugby side Lions in 2012 and, after three years of residency, qualified to play for Scotland in time for the 2015 World Cup.
Number eight Strauss has been sidelined by a kidney injury sustained during the Six Nations defeat by France, ruling him out of the rest of the tournament.
"Since coming to the UK I always had it in the back of my mind that I wanted to play in the Premiership," said Strauss.
"I have really enjoyed my time with the Warriors, but I asked my agent to look around for me. He came back with Sale Sharks who have a good name, a good record in the Premiership and looked an attractive proposition."
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.
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Private companies in Oxfordshire may not have enough capacity to employ people who lose their public sector jobs, an economics expert has warned.
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Premiership side Sale Sharks will sign Scotland and Glasgow back row forward Josh Strauss on a three-year deal from next season.