Dominic Calvert-Lewin scored his first since October from the 15th-minute penalty to put Everton ahead but Leicester equalized seven minutes later through Caglar Soyoncu.
Jamie Vardy brilliantly completed an attack to put Leicester ahead in the 33rd minute.
At the end of the first-half madness, Everton missed two goals, and Leicester playmaker James Madison missed a penalty, which was saved by Jordan Pickford.
Alex Iwobi equalized for Everton 10 minutes into the second half and both sides looked for the winning goal which would have given their survival hopes a boost.
In the end, each side settled for a point, which was more welcome than Leicester, who climbed out of the relegation zone with 30 points from 34 games.
Everton, who have not left the top flight for 69 years but are increasingly likely to be relegated, remain second to last with 29 points from 34 appearances.
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