Promoted to the competition's top tier following their group victory last year, the Swedes frustrated the world champions for the opening 40 minutes until Mbappe proved to be the difference.
The 21-year-old striker got a fortunate bounce off a defender's legs as he danced through the right side of the Swedish defence before beating goalkeeper Robin Olsen from a very tight angle at his near post.
With the clock ticking up towards 90 minutes, Sweden winger Emil Forsberg forced a fine save from Hugo Lloris and Marcus Berg thumped a shot that was deflected wide for a corner that the Swedes could not capitalise on as France held firm.
However, Antoine Griezmann fired the spot kick over the bar with the final kick of the game. Despite France's attack failing to fire, coach Didier Deschamps declared himself satisfied with the result.
"Not everything was perfect. It's a new system that worked pretty well, [but] we could have done better in attacking terms," he told French TV channel M6.
Mbappe was pleased with his coach's tactical switch.
"I like being closer to goal in this system, it makes me happy. I think its a system that suits the characteristics of other players in the team too, and we'll work on it to make it work," he said.