The Premier League outfit led through David Luiz and Eden Hazard, but Aleksandar Kolarov and a double from Edin Dzeko - the Bosnian's first goals against Chelsea in 10 games - turned the match around in West London.
However Hazard's header 15 minutes from time ensured that Chelsea remained at the top of Group C, and with a five-point cushion over third-place Atletico Madrid.
Within five minutes of his return from a hamstring injury, Alvaro Morata should have given Chelsea the lead after being found by Hazard inside the penalty area, but his first-time effort went straight at goalkeeper Alisson.
Up the other end, Diego Perotti was proving a threat to the Blues backline and after drifting inside onto his right foot, the attacker drilled a fizzing effort marginally over the crossbar.
However, Chelsea would open the scoring in the 11th minute through Luiz, who capitalised on his own poor pass to curl a delightful strike into the bottom corner from 25 yards out.
The goal should have led to Chelsea taking control but after Hazard had dragged a low attempt wide from distance, it was the visitors who dominated with the Premier League side being indebted to Thibaut Courtois.
The Belgian made two fairly comfortable saves to deny Perotti, but Radja Nainggolan should have equalised for Roma after being played in on the right-hand side of the area, only for Courtois to prevent his shot finding the near corner.
Against the run of play, Chelsea found a second through Hazard, who netted his first goal of the season after being in the right place to convert Morata's cross at the back post.
Soon after the restart, Roma began to reassert their dominance on the game without creating the opportunities which were going to trouble Courtois.
However, after Luiz was substituted - presumably as a result of a calf injury sustained earlier in the game - Chelsea began to capitulate, although there was little anyone could do to prevent Dzeko's stupendous equaliser.
Roma were very much in the ascendancy and it came as no surprise when the comeback was complete with 20 minutes remaining, with Dzeko glancing a Kolarov free kick into the far corner from close range.
In the space of 33 minutes, Chelsea had blown a two-goal lead but it took the Blues just five minutes to get back on level terms as Hazard netted his second.
Pedro's cross from the right found his teammate on the penalty spot, and the playmaker's header was flicked delicately into the far corner, leaving Alisson with no chance.
With 10 minutes left, Dzeko should have completed his hat-trick - only to head wide from no more than six yards out - while Roma old boy Antonio Rudiger and Gary Cahill both wasted headed chances for Chelsea.
However, a point apiece was a fair result, and one that does not necessarily hinder either team after Atletico Madrid's surprise draw against Qarabag.