Antonio Rudiger's header and Willian's stunning stoppage-time strike secured Chelsea a place in the EFL Cup quarter-finals as David Unsworth's first game in temporary charge of Everton ended in a 2-1 defeat.
Ronald Koeman's 16-month stint in charge of Everton was brought to a halt on Monday, following a humiliating 5-2 home defeat to Arsenal that left them 18th in Premier League.
However, a rejuvenated second-half display showed Everton's quality - with a moment of magic from Willian in the 92nd minute ensuring that Dominic Calvert-Lewin's effort soon after was a mere consolation.
Rudiger's superb 26th-minute effort, the defender's first goal for the Premier League champions, looked to have taken the spirit out of the Toffees, but Unsworth's side responded superbly and dominated in the second half - Willy Caballero pulling off a string of saves to preserve Chelsea's lead.
Ademola Lookman went close with a blistering strike against the bar just before Chelsea wrapped up the win late on.
Ultimately, though, it is now just two wins in 14 matches across all competitions for managerless Everton, who face fellow strugglers Leicester City next time out.
1 - Everton have only kept one clean sheet in their last 14 games in all competitions (v Sunderland in the League Cup). Wounded. — OptaJoe (@OptaJoe) October 25, 2017
Rudiger opened the scoring with the game's first moment of quality as, unmarked at the back post, he powered a brilliant header home from Charly Musonda's cross.
Michy Batshuayi managed to get a shot off on Chelsea's next meaningful venture forward, Jordan Pickford equal to the Belgium striker's low strike.
Devoid of confidence up top, Everton could have conceded again prior to the interval, and were fortunate that Davide Zappacosta could only hit the side-netting after Chelsea capitalised on Ashley Williams' error.
Impressing on his senior debut in Everton's midfield, Beni Baningime spurred the Toffees into life after the restart with a crunching challenge on Ethan Ampadu - Jonjoe Kenny firing wide from the resulting counter.
Everton pressed on and a star-jumping Caballero was out quickly to deny Wayne Rooney in the six-yard box before Aaron Lennon twice went close.
Caballero's fine form continued as Everton piled on the pressure, Chelsea's stand-in goalkeeper keeping out Phil Jagielka's header before denying Kevin Mirallas.
Batshuayi should have wrapped up the victory with just over 10 minutes remaining after latching onto Rooney's sloppy pass, but a poor touch took the ball wide of the post.
Substitute Lookman almost made Chelsea pay with a scorching hit from distance that clattered the woodwork, only for Willian to do the damage at the other end.
Having combined with Fabregas, the Brazil winger cut inside from the left to plant a wonderful curling effort in off the far post, giving Pickford no chance.
There was to be late drama as Calvert-Lewin tucked home from close-range, but it was too little too late as Everton slumped to a third straight defeat.
Key Opta stats:
- 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 reached the last eight on more occasions.
- Chelsea haven't lost a home game against Everton in all competitions since November 1994 – remaining unbeaten in their last 28 (W17 D11).
- Everton have only kept one clean sheet in their last 14 games in all competitions (v Sunderland in the League Cup third round).
- Willian's goal was his first of the season for Chelsea, and ended a run of 14 games in all competitions without scoring (since May v Sunderland).
- 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.