Iceland was 2-0 up with five minutes left at the Roudourou stadium in Brittany and looked set to hand France its first loss since winning the World Cup in July.
But Mbappe, who went on as a substitute for the last half an hour, was the home team's savior.
The Paris Saint-Germain forward helped to pull a goal back when he fired in from a tight angle and the ball took a deflection off Holmar Eyjolfsson to go in for an own goal in the 85th minute.
Mbappe then levelled with a penalty awarded for handball five minutes later.
"We reacted well after a bad start against a well organized team," France coach Didier Deschamps said. "It's good for Kylian, he brought his speed to us."
Deschamps rested several regulars to keep them fit ahead of next week's UEFA Nations League against Germany at the Stade de France.
Without Blaise Matuidi and N'Golo Kante, France were lacking in midfield, while Olivier Giroud was subdued up front.
Mbappe was meant to start the game but a minor thigh injury prompted Deschamps to leave him on the bench, allowing Giroud to partner Antoine Griezmann.
France lacked a cutting edge in the first half despite monopolizing possession.
Unmarked on the edge of the area, Paul Pogba fluffed a shot in the 17th minute. Griezmann also headed a cross from Lucas Digne just wide and Florian Thauvin fired a shot over from Ousmane Dembele's precise pass.
France was made to pay when Presnel Kimpembe was dispossessed near the corner flag by Alfred Finnbogason and went down holding his ankle.
The Iceland forward carried on and sent a low cross to Birkir Bjarnasson, who beat keeper Hugo Lloris with a low shot to claim his 10th international goal in the 31st minute.
France should have gone two down but Lloris made a superb triple stop in the 37th minute.
The hosts pushed hard for an equalizer and almost got it in the 54th minute when Griezmann forced keeper Runar Runarsson to stretch and save a header that looked bound for the top corner.
France was then caught cold from a corner in the 58th as Kari Arnason lost his marker at the near post and doubled Iceland's lead with a looping header that went in off the bar.
Mbappe had the last word, though, setting up the first France strike before bagging his 10th goal in 25 internationals.