Carlo Ancelotti’s side won an entertaining game at the Santiago Bernabeu to put pressure on their bitter rivals, who host Girona in a Catalan derby on Sunday.
Mbappe dinked Los Blancos ahead from the penalty spot but goals from Diego Garcia and Dani Raba sent the minnows ahead before the break.
Jude Bellingham levelled for Real early in the second half before Mbappe whipped home a freekick to claim three points for the champions.
Barcelona and Real Madrid are six clear of third-place Atletico Madrid, who drew at Espanyol earlier today.
Ancelotti named Vinicius Junior and Rodrygo Goes on the bench after they were in action for Brazil on Tuesday, starting Mbappe with Brahim Diaz in attack.
After some good defending and goalkeeper Marko Dmitrovic helped Leganes keep Madrid at bay, the hosts won a dubious penalty.
Oscar Rodriguez tangled legs with midfielder Arda Guler and it was deemed strong enough contact for the referee to point to the spot.
Mbappe executed a perfect Panenka to send Los Blancos ahead, although their lead only lasted a few seconds.
Leganes levelled immediately, with Fran Garcia playing the visitors onside and Diego Garcia netting from close range.
Diaz came close at the other end but then gave the ball away in the build-up to Leganes’ second.
The visitors worked the ball out to Rodriguez on the left, who cut it back for Raba to finish.
Real came out strongly at the start of the second half and Bellingham soon struck his eighth league goal.
The England international followed up to score after Diaz’s shot was blocked on the line, with the ball looping into the air and falling nicely for Bellingham after it bounced off the crossbar.
Diaz struck the post with a low effort after Mbappe teed him up as Real looked to go back in front.
Ancelotti sent on Vinicius and Rodrygo to help complete Real’s comeback.
Eventually Mbappe netted his 22nd league goal of the season with a freekick to put Real Madrid ahead after 76 minutes.
The French striker trails La Liga’s top scorer, Barcelona hitman Robert Lewandowski, by one, and has 33 across all competitions in 44 appearances.
Former Barcelona forward Munir El Haddadi came close to an equaliser for Leganes but nodded wide in the final stages.