Chelsea booked their place in the last eight of the Carabao Cup after a hard-fought 2-1 win over Everton at Stamford Bridge.
Antonio Rudiger (26) scored his first goal for the club since joining from Roma as he rose unmarked at the far post to meet Charly Musonda's short corner.
Everton, who had caretaker boss David Unsworth in the dugout, had their opportunities to equalise in the second period when Ademola Lookman's shot skimmed the crossbar, before Willian scored a brilliant second in stoppage time (90+2).
Chelsea: Caballero (6), Rudiger (7), Christensen (6), Cahill (6), Zappacosta (7), Drinkwater (6), Ampadou (7), Kenedy (7), Willian (8), Batshuayi (6), Musonda (7)
Subs: Fabregas (6), Pedro (6), Morata (6)
Everton: Pickford (6), Baines (6), Williams (6), Jagielka (7), Kenny (7), Baningime (7), McCarthy (6), Davies (7), Lennon (6), Mirallas (6), Rooney (6)
Subs: Lookman (7), Calvert-Lewin (6), Niasse (6)
Man of the match: Willian (Chelsea)
Dominic Calvert-Lewin (90+4) pounced to bundle the ball over the line moments later but it proved nothing more than a consolation as Antonio Conte's side booked their place in the quarter-finals.
Everton began the game on the front foot with Unsworth, on the touchline for the first time since becoming interim manager following the sacking of Ronald Koeman, making his presence felt.
Despite the away side showing good aggression, it was Chelsea who produced the one moment of class in an uneventful opening half.
Chelsea made nine changes to the side that beat Watford with Ethan Ampadu and Danny Drinkwater making their first starts for the club. Everton made five changes to the side beaten by Arsenal as Beni Baningime was handed his first senior start while James McCarthy made his first appearance of the season.
Just as in the Premier League encounter between these two sides, Everton were punished for a lapse in concentration as Rudiger peeled off Williams at the far post to meet Musonda's cross to score after 26 minutes.
It was a fine header back across goal which left Pickford with no chance, but the Toffees emerged for the second half with renewed belief.
So dominant in the first 45 minutes, Chelsea afforded their opponents several chances as Wayne Rooney was denied from close range by Willy Caballero following a knock-down by Kevin Mirallas.
Caballero was called upon twice more in quick succession to deny Mirallas before a slack back-pass from Rooney nearly allowed Michy Batshuayi to score a second before a desperate last-ditch clearance from Phil Jagielka.
Lookman looked bright off the bench for the visitors and the England Under 21 international nearly levelled when his fierce left-foot shot clipped the bar, before Willian appeared to have put the game to bed.
The Brazilian played a sharp one-two with Cesc Fabregas before firing an unstoppable shot low to Pickford's left which flew in off the post.
Calvert-Lewin pulled a goal back deep into stoppage time, but it is Chelsea who enter the quarter-final draw on Thursday afternoon.
The Blues have reached the League Cup quarter-final for the 21st time in their history - only Arsenal (26), Spurs (23) and Liverpool (22) have achieved it on more occasions.
Everton have only kept one clean sheet in their last 14 games in all competitions (v Sunderland in the League Cup third round).
Ethan Ampadu (17y 40d) became the first player born in 2000 to start for Chelsea in any competition.
Chelsea scored with two of their three shots on target in this game, while Willian's goal was their only one of the second half.
Antonio Conte: "I am very happy because tonight I saw lots of positive things, above all the performance of the young players: Musonda, Kenedy and Ampadu. Don't forget Ampadu is only 17.
"They played a good game. It is important to trust these players. Tonight, Michy (Batshuayi) didn't score but he played well, a good performance.
"I think Willy (Caballero) played very well but if I have to find a negative thing about tonight, we conceded in the last minute."
David Unsworth: "It was a totally committed performance. We don't like losing so we're not happy we've lost the game, but I thought the players were outstanding.
"If we get that every week, we'll be absolutely fine. We had a game plan for an hour, but we knew we had to make a change because James McCarthy was always going to come off.
"The plan is to go and win every game, but I wanted a performance as well. The club got that so it's something to build on and take into the Leicester game."
Man of the match - Willian
The one Chelsea player who didn't allow his performance to drop in the second period, Willian produced a moment of sheer class to steer his side into the hat for the last eight.
The Brazilian received a short corner off Cesc Fabregas before losing Tom Davies with the slip of the shoulder and running onto a ball returned to him by the Spaniard and firing a first-time shot into the bottom corner.
The sublime strike was his first goal of the season for Chelsea, and ended a run of 14 games in all competitions without scoring (since May v Sunderland).
Chelsea now prepare for the trip to south coast to face Bournemouth in the Premier League on Saturday evening while Everton visit Leicester at the King Power Stadium as part of Nissan Super Sunday.