The Guinea-Bissau forward produced a clinical finish at an empty Nou Camp before Lionel Messi doubled the lead from the penalty spot.
Messi’s 699th goal for club and country came after the Argentina forward was fouled by Ruben Perez.
Barcelona are five points clear of nearest rivals Real Madrid.
However, Real will cut the lead to two points if they beat Valencia at home on Thursday (21:00 BST).
At 16 years and 298 days, Fati became the second-youngest first-team player in Barcelona history when he made his debut in a 5-2 win over Real Betis on 25 August.
Having moved to Spain at the age of six with his family, he has been with Barcelona’s youth academy since 2012.
The exciting youngster produced a composed finish to put Barca in control before Messi doubled the lead in the first match played at the Nou Camp since La Liga resumed after it was suspended due to the coronavirus pandemic.
Leganes are bottom of the table, three points from safety with nine games left.