Conor Gallagher opened the scoring by lifting a placed shot into the roof of the net. Nicolas Jackson then left Villa Park shellshocked by heading in Malo Gusto's whipped cross.
Enzo Fernandez added real gloss to proceedings by curling a 25-yard free-kick beyond his international team-mate Emi Martinez, with Moussa Diaby scoring a consolation goal for Villa late on.
Victory will relieve pressure on a club whose away fans began chanting for Roman Abramovich but ended the match celebrating with their under-fire players and manager.
The Blues will face Championship contenders Leeds at Stamford Bridge in the last 16.
Standard Sport’s Nizaar Kinsella was at Villa Park to rate the Chelsea players.
Djordje Petrovic - 7
The Serb was busy once again but was a truly a commanding presence. There was little he could have done about Diaby's late strike.
Malo Gusto - 8
A wonderful whipped cross capped off an excellent defensive display to put a chance on a plate for Jackson to score.
Axel Disasi - 7
Chelsea's enforcer almost man-marked the dangerous Ollie Watkins entirely out of the game.
Benoit Badiashile - 6
The Frenchman made a bad mistake that went unpunished early on, losing the ball in the box to Leon Bailey, but he settled as the game wore on. Went off injured late.
Ben Chilwell - 7
The vice-captain led by example and provided wonderful balance down the left-hand side.
Enzo Fernandez - 9
A composed display in midfield was capped off with a sensational free-kick, scored from over 25 yards out in the second half.
Moises Caicedo - 6
His mistake early on led to a big Bailey chance but he improved as the match wore on. The midfielder unfortunately got injured in the second half.
Conor Gallagher - 7
Gallagher got his much-needed first goal of the season by lifting a placed finish into the roof of the net in the 11th minute.
Cole Palmer - 6
Chelsea's top scorer had the first half-chance but snatched at it. He played as the central striker but was not the star of the show.
Noni Madueke - 7
His cross set up Palmer's first effort before a good lay-off set up Gallagher's goal. He then produced a Gareth Bale-like half-of-the-pitch dribble at the end of the first half.
Nicolas Jackson - 7
Jackson played on the left wing and his running power helped set up Gallagher's opener.
Raheem Sterling (for Madueke, 75’) - 6
The winger was brought on midway through the second half but didn't make a major impression.
Thiago Silva (for Palmer, 81’) - 6
Chelsea switched to a back three when Silva came on but he had little to do.
Christopher Nkunku (for Jackson, 87’) - N/A
Alfie Gilchrist (for Badiashile, 87’) - N/A
Unused subs: Bergstrom, Bettinelli, Samuels-Smith, Chukwuemeka, Mudryk.