Paris Saint-Germain attacker Mbappe converted France's only shot on target in the 41st minute of the Group A3 match, threading the ball through the narrowest of gaps between Sweden goalkeeper Robin Olsen and the near post.
The 21-year-old's goal was a bitter blow for the hosts, who earned promotion from League B in 2018-19, given they had the better of the chances up until that point.
Sweden made a late push for an equaliser before Victor Lindelof was caught out by the pace of substitute Anthony Martial, but Griezmann blazed the resulting penalty over with the final kick of the game.
France lacked cohesion in their 3-4-3 formation and nearly fell behind in the 15th minute when Mikael Lustig's venomous 30-yard effort flashed past the post.
Hugo Lloris produced a brilliant save to keep Marcus Berg at bay after a great pass from Albin Ekdal, while Raphael Varane – who was caught out by the Sampdoria midfielder's ball over the top – ensured Emil Forsberg's cross on the follow-up did not find a team-mate.
France took the lead three minutes later, with Mbappe making the most of a fortuitous bounce by tucking home from a tight angle.
Olivier Giroud nodded a cross from Lucas Digne narrowly wide in the 54th minute but Les Bleus struggled to continue to apply pressure.
Martial replaced Mbappe - who took a blow to his right ankle - for his first France cap since March 2018 and, after Leo Dubois inadvertently hit the crossbar, Forsberg and Berg were unable to take chances to equalise.
Sweden's attempts to restore parity left them short at the back and although Lindelof pushed Martial - his club team-mate at Manchester United - over in the box, Griezmann was unable to convert from the spot.
What does it mean? France have food for thought
On paper the front three of Mbappe, Giroud and Griezmann looked like it would cause Sweden no end of problems, but they did not quite click on Saturday.
Griezmann and Mbappe regularly dropped deep to get involved in the play and Giroud was consequently left isolated, leading to a lack of link-up play between the trio.
Despite only being on the pitch for 13 minutes, Martial was able to stake a claim for a greater role in their next fixture.
Mbappe maintains form
There were not many chances for France, but Mbappe made sure their first opening counted. He has now been involved in seven goals in his past seven internationals (four goals, three assists).
Griezmann comes up short
In contrast to Mbappe, Griezmann proved wasteful when opportunities came his way. He missed the target at the end of a quick break in the 76th minute and produced an utterly woeful penalty in the last action of the game – something that will no doubt stick in his memory.