The hosts, backed by many as favorites for the crown, once against failed to live up to the pre-tournament hype in a drab goalless stalemate.
Switzerland, knowing a point would be enough to see it qualify as runner-up, was content to sit back and soak up French pressure for much of the 90 minutes.
France, likewise, knew a draw would guarantee it top spot and rarely committed numbers in attack, not wanting to be caught out on the counter.
Didier Deschamps' side needed a late stroke of genius from Dimitri Payet to win its opener against Romania, while two injury times goals were required to defeat Albania.
There was an expectation that the team would finally start to show its title credentials and the opening 15 minutes against Switzerland certainly pointed to that.
Just after the 10 minute mark, France's golden boy Paul Pogba curled a shot towards the top corner that Swiss keeper Yann Sommer clumsily fumbled onto the top of the crossbar.
A minute later, Pogba was this time met by Sommer in top form. Kinglsey Coman set up his former Juventus teammate, whose fierce half-volley was expertly tipped wide by the Switzerland No. 1.
The frantic opening quarter hour ended with the woodwork once again coming to Switzerland's rescue. After a driving run, Pogba unleashed a ferocious left foot shot clipped the top of the bar with Sommer helpless.
While the first half ended with somewhat of a whimper, the second began at breakneck speed.
A slick one-two between Antoine Griezmann and Pogba resulted in the former firing a shot towards the top corner, but Sommer once again palmed the ball away.
With 15 minutes left to play, Newcastle midfielder Moussa Sissoko surged down the right wing and delivered a pinpoint cross to Payet at the far post, but France's savior could only watch as his volleyed effort hit the underside of the crossbar and bounce away.
The stadium then held its breath expectantly as Payet stepped up to a 20-yard free kick, but let out a disappointed sigh as the West Ham man's effort hit a teammate and bobbled out for a goal kick.
Switzerland had a late shout for a penalty turned down but, in truth, both sides will leave content with their pre-tournament ambitions for the group stages fulfilled.
Meanwhile, in Group A's second game -- being played simultaneously -- Albania created history by scoring its first ever goal at a major tournament as it recorded a first ever major tournament win.
Armando Sadiku headed Ledian Memushaj's cross past the wandering Romanian keeper Ciprian Tatarusanu to spark jubilant scenes among the hoards of traveling Albania fans.
The famous win ensures the country finishes third and now has an anxious wait to see if it will be enough to qualify as one of the four best third-placed teams.