Chelsea became the first English team to lose a Champions League game this season as they were well beaten by a clinical Roma.
The Blues would have progressed with a victory, but trailed within 39 seconds when Edin Dzeko flicked Aleksandar Kolarov's cross down to Stephan El Shaarawy, who smashed home from 25 yards.
Alvaro Morata missed a huge chance to equalise from close range, and El Shaarawy made it 2-0 when ex-Roma defender Antonio Rudiger failed to deal with Radja Nainggolan's long ball forward.
Chelsea had more shots - 14 to 13 - but were three behind when Diego Perotti cracked home from 20 yards after Blues defenders failed to close him down.
Roma should have led 4-0 when Chelsea's defenders all ran to Dzeko, who managed to square to a totally unmarked Perotti, but he fired over when he should have finished.
The Blues were hugely fortunate that Atletico Madrid's failure to beat Qarabag - their game in the Spanish capital ended 1-1 - means Chelsea are still likely to qualify for the last 16 from Group C.
Antonio Conte's side are four points above third-placed Atletico - and a win in their next Champions League game at Qarabag (17:00 GMT on Wednesday, 22 November) will take them through.
Roma, who are fifth in Serie A, now lead the group by a point.
Chelsea get it wrong at the back - again and again
Chelsea had hoped midfielder N'Golo Kante would be able to return from the hamstring injury which has kept him out for a month, but he did not feel fully fit and was left out.
And how the Blues missed him. David Luiz started in midfield in the reverse game - a 3-3 draw at Stamford Bridge two weeks ago - but played in defence at the Stadio Olimpico. Centre-back Gary Cahill replaced Davide Zappacosta from Saturday's win over Bournemouth, meaning a reshuffle with Cesar Azpilicueta moving to the right.
They actually should have been 1-0 up within 21 seconds - when Pedro shot straight at Alisson Becker - before El Shaarawy scored the sixth quickest Champions League goal of the past decade, following good work from ex-City pair Kolarov and Dzeko.
Chelsea had plenty of shots - but rarely looked like scoring, with the lively Hazard forcing two saves and shooting wide. Morata also missed a sitter and Marcos Alonso forced a good diving save with an effort from the corner of the box.
But the Blues were error-ridden at the back. They should have conceded when Luiz made a mistake on the halfway line and El Shaarawy played in Dzeko, whose shot was saved.
They trailed 2-0 when Rudiger, a summer signing from the Italian side, let Nainggolan's pass bounce and El Shaarawy took advantage to score past the onrushing Thibaut Courtois.
Things did not improve when Antonio Conte brought on Willian for captain Cahill, moving Pedro to the right - and the Spaniard let Perotti go past him with no other defenders shutting the Argentine down before he smashed in the third.
They were lucky not to concede again with defending resembling football seen on five-a-side pitches - or school yards - across the world. All three Chelsea centre-backs ran to Dzeko - with two men free in the centre but Perotti got it wrong.
Courtois denied Kostas Manolas' header with a good reaction save as Chelsea were just glad to get off the pitch at the end.
Morata struggling for confidenceAlvaro Morata only had 26 touches - missing two chances - before being withdrawn
Chelsea striker Morata had another disappointing evening in front of goal.
The 25-year-old Spain forward, a £60m summer signing from Real Madrid, had scored seven goals in his first eight games in English football.
But it is now six games in a row without scoring and he was replaced by Michy Batshuayi with 15 minutes left.
At 1-0, he missed a glorious chance to score. Kolarov's clearance hit Pedro and landed at Morata's feet six yards out. The Spaniard was not expecting the ball and blazed well over.
He only had two shots on a quiet night - dragging a shot wide from just outside the box after a one-two with Hazard.
Former England defender Phil Neville - at the Stadio Olimpico for BBC Radio 5 live - said: "Morata missed two really good chances at Bournemouth on the weekend and he's not scored for five games [before today].
"I just wonder if his confidence has been knocked. He had to score that."
Could Morata be in danger of losing his place for Sunday's Premier League game against Manchester United at 16:30 GMT? Roma go to Fiorentina at 14:00 GMT that day.
Man of the match - Aleksandar Kolarov (Roma)Ex-Manchester City player Aleksandar Kolarov was impressive at left-back - having a hand in El Shaarawy's opener and assisting Diego Perotti's third. He had 13 more touches than anyone else (92)
Match stats - Fabregas reaches landmark
- The other 16 games involving English clubs in this season's Champions League have seen them win 12 and draw four.
- Chelsea are now winless in their past six away games against Italian clubs in the Champions League (D1 L5).
- This was only the third time that Chelsea have lost a Champions League game by a margin of three or more goals. The previous two were 5-1 at Barcelona in April 2000 and 3-0 at Juventus in November 2012.
- Stephan El Shaarawy's opening goal was the first time that Chelsea have conceded in the opening minute since 1 December 2013 when Jay Rodriguez netted after 13 seconds for Southampton.
- This goal was also the quickest that Chelsea have ever conceded in the Champions League and the quickest scored in the 2017-18 competition so far.
- Cesc Fabregas made his 100th Champions League appearance in this match, becoming the fifth youngest to reach this landmark (30 years, 180 days).