Mbappe picked up the ball wide of the goal in the 41st minute, squeezed past two defenders, waited for goalkeeper Robin Olsen to go down and then clipped the ball just inside the post from the narrowest of angles.
The strike was enough to give France victory in a cagey opener in Group 3 of the elite League A in the competition.
"It was a committed match, not a great match," said France manager Didier Deschamps. "It was a good back-to-school game, considering the fitness levels."
Mbappe agreed that fitness, after the disruption caused by the coronavirus pandemic, explained the lacklustre game.
"We are not all at the same level," Mbappe said.
Sweden managed just three strikes on target in the 90 minutes, though one of those was a deflection off French defender Dayot Upamecano, making his international debut, that Hugo Lloris saved.
Antoine Griezmann had a chance to double the French lead with the last kick of the game after Anthony Martial was fouled in the box, but the Barcelona attacker blasted the penalty high.
"It was a complicated match," said Mbappe. "We competed well, it wasn't easy."
Deschamps has adopted a new system with two strikers.
"It's clear that I'm closer to the goal," said Mbappe. "I like that."
Captain Lloris also focused on the formation and a lack of fitness.
"In the second half, we lost intensity, I found the Swedes very good physically," said the goalie, adding that the team were still learning the new system.
"We're not going to adapt overnight, we had very few days to prepare for it, we have to work on it, we have to give it time."
For France's first match since they beat Albania last November, Deschamps recalled Juventus midfielder Adrien Rabiot for his first appearance since he refused to serve as a standby for the 2018 World Cup.
France face Croatia in a rematch of the 2018 World Cup final at the Stade de France in Paris on Tuesday while Sweden take on the fourth team in the group, Portugal, winners of the inaugural Nations League, in Solna.
France travelled to sweden with just two goalkeepers after Steve Mandanda of Marseille was sent home because he was deemed to have failed a Covid-19 test and there was no time to arrange tests for a replacement.
After Saturday's victory, Deschamps said he had called up Benoit Costil of Bordeaux, who will join the squad on Sunday.