The visitors opened the scoring early on through Chris Long’s neat finish, but March’s Barclays U21 Premier League Player of the Month Harry Panayiotou levelled just 10 minutes later.
Finishing 1-1 in normal time, it was Panayiotou who would grab his second of the game to put City 2-1 up in extra-time, only for Harry Charsley to force the fixture to penalties, ensuring the tie finished 2-2 after 120 minutes.
Ryan Ledson, Kieran Dowell and Liam Walsh converted Everton’s first three penalties, while Joe Dodoo and Ben Chilwell were on target from the spot for Steve Beaglehole’s outfit, before Russell Griffiths denied Simonas Stankevicius for City’s third.
But it was Leicester shot-stopper Adam Smith who proved decisive, saving from Jon Joe Kenny and Tyias Browning in emphatic fashion, allowing Liam Moore to score the winning penalty to send the Foxes into the last-four.
First-teamers Anthony Knockaert and Moore both notched up valuable game time in a fixture that provided City with a second Merseyside opponent in four days, having beaten Liverpool in the league at King Power Stadium on Monday evening.
Everton started with real intent as Long fired his side into the lead from inside the box with just 10 minutes on the clock.
Moore then nodded Dodoo’s teasing cross narrowly wide just moments later, but it would be St. Kitts and Nevis international Panayiotou who would level things up on 20 minutes, converting defender Kieran Kennedy’s knock-down from three yards.
Opposition winger David Henen then tested Smith from the left, but the ‘keeper was wise to the effort, and clawed the ball out of the top corner to keep the score level.
The lively Panayiotou could have edged City ahead just after the half-hour mark, only to head wide from deep inside the visitors’ box from Chilwell’s delivery.
At the other end, Chilwell continued his defensive duties well, blocking Tom Davies’ low drive – goalscorer Long firing inches wide from the rebound.
A looping 20-yard effort from Barmby then brought the first half to a close – the ball nestling harmlessly on top of Griffiths’ net.
A siege on the Toffees’ goal from City then ensued, with Knockaert, Moore and Panayiotou all having chances to put the home side into the lead for the first time in the game.
The impressive Walsh had the best opportunity for David Unsworth’s team in normal time, rattling Smith’s crossbar with a curling free-kick with 20 minutes to go.
Only Griffiths would prevent the Foxes stealing it late on, denying Dodoo one-on-one following a lethal break from the hosts – 1-1 after 90 minutes.
A blistering start to extra-time saw Panayiotou scramble in, again from close range, from Moore’s knock-down to put Leicester into the lead.
But Everton rallied, and just three minutes later, they were on level terms. A well-weighted pass from Dowell found substitute Charsley, who finished low past Smith.
Dowell then dragged a low strike wide from distance, while at the other end, Barmby couldn’t direct his drive on target, meaning penalty kicks would decide the tie.
Everton got their noses in front, dispatching all three of their first penalties, while Lithuania international Stankevicius saw his effort from 12 yards tipped wide by Griffiths.
But after making a number of key saves in the game, it was City goalkeeper Smith who would prove to be the hero of the fixture.
First, the shot-stopper palmed Kenny’s effort away before denying Browning from the spot either side of Barmby’s excellent penalty.
That left Moore the chance to win the game for City, which he duly took, side-footing past Griffiths to send his side into the semi-finals of the Premier League international Cup.
The draw for the last-four will take place once all of the quarter-final fixtures have been played, with the last set for Thursday evening. Fulham, Porto, Manchester City or Celtic are the potential opponents.
Next, the Foxes entertain Manchester United in the U21 Premier League at King Power Stadium on Monday evening (7pm kick-off) – a crucial tie which could see City move top of the table.
CITY: Smith, Sesay (c), Chilwell, Scott (Miles 91), Kennedy, Moore, Knockaert (Rowley 88), Barmby, Stankevicius, Panayiotou, Dodoo
SUBS NOT USED: Kipre, Maddison, Bailey
GOALS: Panayiotou 20, 94
EVERTON: Griffiths, Kenny (c), Robinson, Ledson, Browning, Galloway, Dowell, Walsh, Long (Dyson 62), Davies (Connolly 66), Henen (Charsley 83)
SUBS NOT USED: Hunt, Donohue, Langton, Byrne
GOALS: Long 10, Charsley 97
REFEREE: Andy Hendley