Mbappé - one of the stars of the France World Cup winning squad - had started the match at the Municipal du Roudourou on the bench.
But with the hosts trailing 2-0 he came on in place of Antoine Griezmann. In the 86th minute his shot at goal was saved by the Iceland goalkeeper Runar Alex Runarsson but the ball hit defender Holmar Orn Eyjolfsson and trickled over the line.
Mbappé equalised in the 90th minute after Kolbeinn Sigthorsson handled the ball in the box.
"The team saw we needed to do something else. I bring what I can do," Mbappe told the TV channel TF1.
"We saw the fans were completely behind us even if it wasn't a big stadium. We wanted to do everything for the people who were there.
We got the draw, we didn't lose, but we're still a bit annoyed."
France coach Didier Deschamps fielded six of the players that began the World Cup final against Croatia in July in Moscow.
He gave the Marseille winger Florian Thauvin his first start ahead of Mbappé who scored four goals for his club on 7 October in their 5-0 destrcution of Lyon.
But the Deschamps shake-up failed to reap rewards. Erik Hamren's men took the lead after 30 minutes when Birkir Bjarnason beat the France goalkeeper Hugo Lloris with a low strike.
Kari Arnason doubled the advantage after 58 minutes. But that set the scene for the Mbappé rescue show.
"There are times when these kind of games come up and give you a shock," said France forward Olivier Giroud. "They remind you that you have keep your feet on the ground and do the things that brought you the success in the first place.
"Iceland showed they wanted a good result and they defended well when they were 2-0 up but we did well to come back and get the draw."
France will be in action in the Uefa Nations League on 16 October when they play Germany at the Stade de France.