The victory means Everton are only one point behind Foxes in fifth in the Premier League going into their meeting with Arsenal on Saturday.
Story of the match
The Foxes started the game the brightest, pressing their opponents fro the front, but struggling with the final ball.
Youri Tielemans was inches away from getting the Foxes off to the perfect start, but he dragged his delicate effort too wide.
Just as Leicester started to increase their tempo and intensity, Everton broke the deadlock courtesy of a Kasper Schmeichel error.
Richarlison's run inside and effort looked easy save for the Danish goalkeeper, but his save turns the ball into the corner of the net.
The Brazilian's opener is his first Premier League goal since the 2-1 victory over Crystal Palace at the end of September.
Calvert-Lewen and Gylfi Sigurdsson had chances to double their lead, but both failed to connect with the ball correctly from Alex Iwobi's cross.
As the first half grew old, the game stretched, with Dominic Calvert-Lewin heading straight to Schmeichel, before Cengiz Under's low drive was easily collected by Olsen.
Leicester dominated the proceedings in the second half, enjoying spells on the ball but failing to force a save out of Olsen.
Maddison aimlessly fired over after good play from Tielemans and Barnes, but Everton netted their second goal on 72 minutes.
Following a brilliant double save from Schmeichel to deny Yerry Mina and Calvert-Lewin, Holgate was the quickest to react to poke the ball into the empty net.
Leicester had a penalty chalked-off by VAR and saw a Vardy header ruled out for offside in the dying stages as their home woes continue.
Foxes beginning to miss key players
Despite overcoming Brighton & Hove Albion on Sunday, Leicester's issues without their key players continued as they had no answer against Everton.
Christian Fuchs struggled against Iwobi, whilst Wilfred Ndidi's makeshift performance at centre-back, although wasn't shocking, demonstrated the need for the return of Caglar Soyuncu and Ricardo Pereira.
There is no doubt about the quality and depth of the Foxes squad, but with their key players missing, they lack the efficiency to overcome teams like Everton.
Poor home form continues
The defeat means Leicester have fallen to four home defeats this season, equalling last season's home tally.
When the Foxes lifted the Premier League trophy in 2016, their home form was the main factor in their incredible journey, but it now seems to be gone.
Despite winning on Sunday at home, the failure to dominate and win at home will worry Brendan Rodgers.