Two goals in three minutes from James Maddison and Jonny Evans before the half-hour mark had the Foxes in control at Goodison Park, however Tom Davies gave the Toffees hope with a side-footed volley 20 minutes from time. A last-gasp wonder-strike from Leighton Baines brought the scores level and took the game to a penalty shoot-out.
Jamie Vardy would score the winning spot-kick for Brendan Rodgers' side - who would reach their first semi-final since 2000 - in a shoot-out which included two saves from Kasper Schmeichel to help Leicester on their way to the win.
Story of the Match
After a cagey start to the game, Everton almost take the lead after a poor clearance from Leicester keeper Schmeichel, which Richarlison was unable to capitalise on after it had originally deflected off Alex Iwobi's head.
Leicester right-back Ricardo Pereira had enjoyed a lot of joy up against Everton left-back Baines and utilised his pace to help the away side open the scoring. By driving past the former England man towards the touchline, Pereira was able to cut the ball back for an unmarked Maddison, who used the outside of his foot to curl the ball into the bottom left hand corner past Jordan Pickford.
The Foxes soon doubled their lead just three minutes later as goalscorer Maddison's corner was flicked on by Jamie Vardy to find the onrushing Evans who bundled the ball into the goal unmarked.
The comeback is on
Leicester had chances to put the game to bed in the second-half with Ayoze Perez testing Pickford from distance as well as a curved shot from Mark Albrighton clipping the crossbar and Duncan Ferguson's Everton took advantage.
A good spell of intricate play from the Toffees enabled the Brazilian Richarlison to spin away from Ben Chilwell and deliver an enticing cross for local lad Davies who volleyed past Schmeichel to half the defecit.
As Everton continued to press forward, their relentless attack finally paid off. As the clock ticked into the 90th minute it was Everton legend Baines who would find the equaliser, striking the ball from 25 yards into the back of the net and sending the home faithful into jubilation.
Penalty joy for Leicester
After both keepers were able to save the opening penalties for either side, it was Chilwell who would break the penalty deadlock, comfortably slotting past his England team-mate.
Schmeichel would then save Baines' spot-kick before Pereira, Demarai Gray and Dominic Calvert-Lewin would all dispatch from the spot.
Leicester knew they needed to score to win and there was no other player they would have rather turned to other than Vardy who rifled his penalty past Pickford to secure Leicester's passage to the Carabao Cup semi-finals where they are set to meet Midlands rivals Aston Villa.
Key Takeaways from the Match
Leicester show mental character
The quarter-finals have proven to be a difficult test in recent times from Leicester and for a brief moment it looked as if they could see the tie slip away from them for a third time.
Rodgers' side however remained calm and bar Maddison's miss, dispatched their penalties confidently to take them one step away from Wembley Stadium.
Foxes need to be more ruthless
Despite the penalty success, the Foxes had plenty of chances to finish the game off in 90 minutes before letting Everton back into the tie.
Dennis Praet, Maddison, Perez and many more all had golden opportunities to extend the lead for the away side and put the result beyond Everton's reach however they were unable to capitalise - allowing the home side back into the game - and make it a nervy end for the travelling Foxes fans.