The Foxes went in to the game having beaten Chelsea 2-1 on Monday night, and continue to shock the world by sitting top of the Premier League table.
Claudio Ranieri's men opened the scoring in the first half when Ramiro Funes Mori fouled Shinji Okazaki inside the area, with Riyad Mahrez scoring from the penalty spot.
However, in-form Romelu Lukaku brought Everton level after Ross Barkley had failed to find the net twice, only for Mahrez to score another penalty after Tim Howard brought Jamie Vardy down in the second half.
Just minutes later, Japanese forward Shinji Okazaki added a third goal to essentially wrap up the win, before Kevin Mirallas snatched a late consolation for Everton.
The result means Leicester have now gone ten games unbeaten, and will be top of the Premier League table at Christmas in a stunning story.
Meanwhile, Everton's six-game unbeaten run is over after today's defeat, meaning they sit in tenth place on 23 points so far this season.