Chelsea seized the advantage when David Luiz curled into the corner from outside the area, although that did not tell the story of a first half in which Roma were frequently on the front foot.
As Chelsea absorbed the pressure, Eden Hazard tucked a shot away at the far post following a counter on 37 minutes for 2-0, but Roma left-back Aleksander Kolarov reduced the deficit soon afterward, pushing the ball past Cesar Azpilicueta before unleashing a powerful shot that deflected past Thibaut Courtois.
Edin Dzeko produced the goal of the game on 64 minutes, connecting perfectly with Federico Fazio's cross to volley home for 2-2, and the former Manchester City striker then put Roma ahead when he headed home Kolarov's free kick.
However, Chelsea found an equaliser when Hazard headed in a Pedro cross to set up a frantic finale, but neither side could produce a winner.
The draw keeps the Premier League champions two points clear of Roma at the top of the group.
The result means Qarabag, who lost Dino Ndlovu to a second yellow card on 75 minutes, become the first Azerbaijani club to pick up a point in the Champions League.
Atletico have taken just two points from their opening three games.
Manchester United battled to a 1-0 victory away to Benfica to make it three wins from three.
United struggled against a defensive Benfica side that had yet to take a point in Group A, but they finally found the breakthrough on 65 minutes when young Benfica keeper Mile Svilar allowed the ball to cross the line when saving Marcus Rashford's free kick.
Benfica defender Luisao was sent off in injury time for a foul on Scott McTominay.
Taulant Xhaka scored Basel's opener, racing through before unleashing a powerful shot from distance on 29 minutes, and the Swiss side saw a second ruled out for offside early in the second half.
Basel made the game safe in the 90th minute when Dimitri Oberlin slotted home.
Lionel Messi scored his 100th goal in European competition as Barcelona maintained their perfect record in Group D with a 3-1 victory over Olympiakos, despite the dismissal of Gerard Pique in the 43rd minute.
Young centre-back Dimitrios Nikolaou diverted Gerard Deulofeu's cross into his own goal on 18 minutes, and Barca thought they had a second before half-time when Pique diverted home from close range, but the Spain international was deemed to have used his hand and picked up a second yellow card.
Messi produced his landmark goal with a 61st-minute free kick, and the Argentina star delivered a fine assist for Lucas Digne to make it three soon afterwards.
Nikolaou headed home a late consolation for the Greek side.
Juventus came from behind to beat Sporting 2-1 in Turin.
Sporting took the lead when Gelson Martins' effort deflected back off goalkeeper Gianluigi Buffon and into Alex Sandro for an own goal, but a superb Miralem Pjanic free kick had Juve level at the break.
Juve went ahead through Mario Mandzukic's fine backpost header on 84 minutes as the Italian champions moved three points clear of their Group D opponents.
PSG went ahead inside the opening three minutes when Kylian Mbappe fired home from a tight angle, and while the Belgians responded well, Edinson Cavani headed home a second after Anderlecht keeper Matz Sels failed to deal with a fierce Neymar strike.
Neymar added a third with a low free kick, but Anderlecht were unfortunate not to reduce the arrears when Everton loanee Henry Onyekuru hit the underside of the crossbar.
Angel Di Maria rounded off the win with a neat chip over Sels late on.
The win maintains PSG's perfect record in the group, while Anderlect are yet to get off the mark.
Bayern Munich responded to the defeat against PSG in their last Champions League outing by beating Celtic 3-0 at the Allianz Arena.
Celtic had an early reprieve when a Thiago Alcantara strike was chalked off, but Thomas Muller soon made amends, firing home at the far post after Robert Lewandowski's header had been saved.
Bayern -- who marked Jupp Heynckes' return with a 5-0 win over Freiburg at the weekend -- doubled their lead through Joshua Kimmich's header just before the half-hour mark.
Mats Hummels headed home from a corner on 51 minutes to extinguish any hope of a Celtic comeback.