Both sides were under pressure to win Sunday’s tie after underwhelming starts to their campaigns which saw their respective managers lose their jobs in recent weeks.
And Leicester’s in-form forward Jamie Vardy got the ball rolling with a tap-in after a fast Foxes break had caught the Toffees cold in the 18th minute.
Demarai Gray, playing just behind Vardy, was the architect of the opener as he surged forward with the ball before playing it out wide to Riyad Mahrez. The Algerian winger sent in a first-time cross across the face of goal with his weaker foot and Vardy arrived at the far post to guide the ball into Jordan Pickford’s net.
Everton, 18th in the Premier League table, tried to respond but soon after were struck by another blow as young right-back Jonjoe Kenny turned the ball into his own net 11 minutes later.
Gray sent in a cross from the left intended for Vardy but Kenny’s attempted clearance backfired spectacularly as a helpless Pickford saw the ball sail past him into the net.
The Toffees, who last recorded a win in September, then had a genuine appeal for a penalty waved away when winger Aaron Lennon was tumbled over by Christiean Fuchs inside the box.
Kevin Mirallas would go close from distance with a curling effort that went just wide as Everton poured forward in a bid to find a goal before the interval. It wasn’t to be, however, and the teams went into the dressing rooms with Leicester holding a comfortable lead.
The second-half saw an improved performance from Everton, but even the half-time introduction of Oumar Niasse wasn’t enough to breach Kasper Schmeichel’s goal.
Half-chances came Leicester’s way as well but the the first-touches of Vicente Iborra and Vardy, respectively, let them down as the Foxes were unable to build on their first-half lead.
With the result, Leicester move up to 11th place in the Premier League standings while Everton remain mired in the relegation zone with a points tally of eight points from 10 matches keeping them a point above 19th placed Bournemouth.
The Foxes travel to unpredictable Stoke on Saturday seeking their third league win on the trot while Everton will try to break out of their funk when they host Watford on Sunday.