The 19-year-old became the youngest France player to score in a major tournament when he fired home from close range in the first half in Yekaterinburg.
Pedro Aquino hit the post for a lively Peru, who are still waiting for their first World Cup win since 1978, and the South American side can no longer qualify from Group C.
The 1998 tournament winners are top of the group after the second round of matches, two points ahead of Denmark and five in front of third-placed Australia, meaning they will now progress.
They face Denmark in their final game to determine who will top the group, while the Danes need just a point to progress.
GOAL VIDEO: Kylian Mbappe (35') - France 1-0 Peru
GOAL! Pogba slides in Giroud whose shot deflects over the goalkeeper and Mbappe is on hand to tap in becoming France's youngest ever goalscorer at a World Cup in the process pic.twitter.com/TGx7b2XZpC
France have kept a clean sheet in each of their last seven World Cup games against South American sides – last conceding in a World Cup quarter-final against Brazil in 1986.
France have qualified for the knockout stages of the World Cup for the sixth time in their last eight appearances in the competition – only failing to in 2002 and 2010.
Peru are just the second South American side to fail to qualify for the World Cup knockout stages in the last three editions alongside Ecuador in 2014.
Peru have now attempted 27 shots without scoring at the 2018 World Cup finals; only Morocco (28) have had more without a goal.
French forward Kylian Mbappé has become the youngest player to ever score a goal for France at a major tournament (19 years 183 days).
France teenager Kylian Mbappe is the first player to score a World Cup goal who was born after France's 1998 triumph in the competition
Fast Results
Scores
Racecards 
