Stephan El Shaarawy scored after 38 seconds and pounced on an Antonio Rudiger error to make it 2-0 after 36 minutes.
Diego Perotti struck a fine third after 63 minutes and there would be no Chelsea comeback this time as Roma moved on to eight points, one more than the Blues.
A win in Antonio Conte’s first match in Italy as Blues boss would have seen Chelsea advance to the knockout stages.
Now Conte’s men need points at Azerbaijan’s Qarabag before the final group clash with Atletico Madrid.
SImon Johnson assesses the key talking points in Rome…
Conte has to take the blame again
Two weeks ago Conte said he was at fault for the 3-3 draw against Roma because he changed formation to have three in midfield.
This time he decided to tinker with his backline and it backfired spectacularly from the outset.
Since Conte switched to three in defence last season, captain Gary Cahill has played at left centre back the vast majority of times.
Meanwhile summer buy Antonio Rudiger is renowned for favouring the right hand side.
For some reason, in what was going to be a very tough assignment, the Chelsea coach switched them round.
It caused confusion from the outset, with Rudiger in particular out of position for Roma’s first and asleep for the second, when he mysteriously allowed the ball to go straight past him.
Arguably even more bizarre was his decision to take off Cahill in the second half and replace him with Willian, moving Pedro to right wing back and Cesar Azpilicueta inside.
Again it didn’t work. Pedro was beaten easily by Diego Perotti before he drilled home the third.
Blues could do with former Man City duo
Chelsea couldn’t cope with Edin Dzeko or Aleksandr Kolarov at home or away as they clearly played with a point to prove against English opposition.
Dzeko left Manchester City in 2015, Kolarov in the summer and they look very impressive in Roma’s colours.
Manchester City may not have wanted to do business with Chelsea for either player, they weren’t even linked as targets, but they are both excelling in areas where the Blues are short of talent.
Marcos Alonso has no real competition at left wing back after Chelsea missed out on Alex Sandro during the last transfer window, while Dzeko is looking a superior striker to both Alvaro Morata and Michy Batshuayi at the moment.
Kolarov and Dzeko combined to set up Stephan El Shaarawy’s opening goal, with the former particularly troublesome all night.
Fabregas won’t have enjoyed his landmark
Fabregas became the fifth youngest player in Champions League history to reach his century of appearances.
Any chance of the Spain international enjoying his achievement was soon lost as Chelsea were woeful at both ends of the pitch.
Fabregas did his best in the first half to try and conjure a goal for the visitors, but his through balls didn’t quite come off.
He faded after the break and to add insult to injury, Fabregas gave the ball away in the build-up to Roma’s third.
Fabregas didn’t see the game out either, coming off in the 71st minute for summer buy Danny Drinkwater.
Forget London traffic – Morata has other worries
On the eve of the game, Chelsea’s record buy admitted that congestion in the capital has been stressing him out.
Morata didn’t seem to be having any issues on or off the pitch at the start of the season, when he was finding the net regularly.
But he has now gone six matches without a goal and is worryingly missing simple chances.
Against Bournemouth on Saturday, he could have had a hat-trick. Here he skied a glorious opportunity over the bar from a few yards out.
The score was still 1-0 at the time and so an equaliser at that point, which Chelsea would have deserved on the balance of play, could have completely altered the end result.
If Chelsea are to bounce back domestically and in Europe from this, Morata needs to rediscover his form fast.
Chelsea’s place in last-16 now under threat
After winning their first two group games, it looked like Conte’s side were going to cruise through to the knock-out stage.
A terrible return of one point in their two games against Roma has changed all that, but the manner of their performances have arguably been more alarming.
From the highs of that win at Atletico Madrid, comes the ignominy of conceding six goals in two matches against the Serie A side.
Many people will have seen the trip to Qarabag in three weeks’ time as a gimme at the start of the campaign, especially after a 6-0 success at home in September.
But Qarabag have shown they are no soft touch with their displays against Atletico Madrid and even they will fancy facing Chelsea on this form.