Antonio Rudiger scored the opening goal of the game in West London, giving Chelsea the lead in an otherwise drab and uneventful first half.
Everton - playing their first match since sacking Ronald Koeman on Monday - pushed for an equaliser in the second half and came close on a number of occasions, but ultimately the hosts got the game's second goal in stoppage time when Willian tucked home a clinical finish.
Dominic Calvert-Lewin quickly restored the slim visiting hopes with an immediate response, but it was too little too late for the Toffees as they crashed out of the competition.
Chelsea handed full debuts to summer signing Danny Drinkwater and 17-year-old Ethan Ampadu in midfield, while the Toffees also had a debutant in the middle of the park with Beni Baningime given his bow for the Toffees.
The swathes of changes from Antonio Conte and caretaker Everton boss David Unsworth saw both sides struggle to get into any sort of rhythm in the opening exchanges, though, and it wasn't until Rudiger's 26th-minute goal that either had any joy in the final third.
The goal was a moment of quality which had hitherto been sorely lacking in the final third, though, as Charly Musonda collected a short corner before swinging a deep cross to the back post for Rudiger, who stretched every sinew to reach the ball and loop an inch-perfect header over Jordan Pickford and into the far corner.
It was Rudiger's first goal for Chelsea since his summer arrival from Roma, but it didn't exactly open the floodgates and clear chances were still scarce as the half ticked by.
Michy Batshuayi - hat-trick hero in the last round and scorer of two goals off the bench at the weekend - sent one tame effort trickling into the arms of Pickford with just under 10 minutes remaining of the half, before Davide Zappacosta slammed his own strike into the side-netting in the closing stages of the opening 45.
Everton went into the break having failed to register a single shot on target, but they should have been ahead 11 minutes into the second half when Kevin Mirallas cushioned a clever header down to Wayne Rooney, whose volley on the turn was thwarted at point-blank range by Willy Caballero.
Caballero almost turned from hero to villain just seconds later when he gifted possession to Aaron Lennon inside his own area, but the winger's heavy first touch allowed the stand-in Chelsea keeper to redeem himself with a goal-saving challenge.
Everton were suddenly in the ascendency having offered nothing going forward up until that point, though, and they came close again on the hour mark when Mirallas's cross was flicked narrowly over the crossbar by Lennon.
Chelsea were hanging on, though, and the previously-quiet Caballero was suddenly overworked as he made two more smart stops in quick succession to deny first Phil Jagielka and then Mirallas.
Everton looked like a different team in the second half and put together a slick passing move to create another sight of goal with less than 15 minutes remaining, but Ademola Lookman could only find the side-netting with his low drive from a tight angle.
Despite Everton's second-half dominance, Chelsea almost killed the game off shortly afterwards when Batshuayi pounced on a wayward back-pass from Rooney and poked the ball beyond Pickford, only for Jagielka to produce a goal-saving last-ditch challenge just as the striker was about to roll it into an empty net.
The Toffees were soon back on the front foot, though, and they came even closer to forcing extra time when Lookman ducked inside onto his left foot before unleashing a fine effort which clipped the top of the crossbar on its way behind.
Any dwindling hopes of forcing a deserved period of extra time seemed to have been extinguished in the second minute of stoppage time when Chelsea doubled their lead through Willian, who collected a short corner before exchanging passes with Cesc Fabregas and picking out the bottom far corner with aplomb.
Everton immediately responded with a quick goal back as Oumar Niasse broke into the box before the ball fell at the feet of Calvert-Lewin, who steered a messy finish past Caballero to reignite the visitors' slim glimmer of hope.
However, there was ultimately no way back for the Toffees as Unsworth's first game in temporary charge ended in defeat, stretching Everton's wait for a Stamford Bridge win to 28 matches across all competitions.
The Merseysiders are also now winless in their last six across all competitions, whereas Chelsea have reached the quarter-finals of this competition for the fifth time in the last seven seasons.