Toffees goalkeeper Nico Hansen saved a first-half penalty from Admiral Muskwe. But the Foxes forward went on to break the deadlock from a 56th-minute corner and Callum Wright, pictured, doubled the lead mere seconds later.
Anthony Gordon scored a late spot-kick for the reigning champions at the other end, but Leicester held on to secure their eighth victory in all competitions and moved up to third with 22 points.
Everton lost for the fourth successive match, staying eighth with 13 points.
Brighton remain sixth with 16 points while Wolves stay 10th with 11 points.
Gabriel Kyeremateng's third-minute strike proved decisive as Stoke won 1-0 at bottom-placed Sunderland, climbing to third.
And a stoppage-time winner from Kees de Boer gave Swansea a 2-1 home win over Norwich City, after Ali Al-Hamadi and Akin Famewo had traded blows for the hosts and visitors respectively.