However, Hazard headed in to prevent Antonio Conte’s side slipping to a third straight loss for the first time under the Italian’s stewardship.
Hazard’s first goal of the season followed David Luiz’s opener but Aleksandar Kolarov halved the deficit before Dzeko’s delightful volley and free header within seven second-half minutes.
Chelsea were second best, but Hazard headed in Pedro’s cross to give the Blues seven points ahead of the trip to the Eternal City on October 31.
The Blues were seeking a response to successive Premier League losses to Manchester City and Crystal Palace which have left the champions nine points adrift ahead of Saturday’s fixture with Watford.
The Blues did not get the win they craved, but will have been boosted by the way they fought back after ceding the initiative to Roma.
Chelsea marked their Champions League return after last season’s European absence with a 6-0 win over Qarabag and followed that up with an impressive display in winning at Atletico Madrid.
A draw in Baku between those sides earlier on Wednesday gave Chelsea added incentive. Beat Roma here and they would have one foot in the knockout stages with three group games to play.
If only it were so simple. Luiz’s deployment in midfield, with N’Golo Kante out injured, provided balance, while striker Alvaro Morata made a welcome return from his own hamstring injury, but Roma were better throughout.
Morata spurned an early chance, shooting straight at Alisson.
Diego Perotti shot over — one of three efforts by the Roma midfielder in the first half — before Luiz struck against the run of play.
The Brazilian roamed forwards and, when his attempted pass to Morata was returned to him off Federico Fazio, he curled the ball in first time from 25 yards.
Roma’s best fell chance fell for Radja Nainggolan, who momentarily escaped Luiz’s attentions and ghosted down the Chelsea left. Thibaut Courtois saved smartly and, soon afterwards, Chelsea doubled their lead.
Hazard fed Morata, whose shot looped up off the outstretched foot of ex-Tottenham defender Fazio and Hazard despatched the ball with ease.
Kolarov took it upon himself to reduce the arrears. The Serbia left-back beat Cesar Azpilicueta before unleashing a fierce shot which deflected off Andreas Christensen and beat Courtois.
Roma continued to pour forwards on the resumption, with one Kolarov cross going just behind Dzeko.
Luiz, who appeared to have been playing with a knock, made way for Pedro and Roma equalised in stunning fashion soon afterwards.
Fazio brought the ball forwards and Dzeko met his lofted pass with a sweetly struck left-footed volley.
The striker’s first career goal against Chelsea was followed by his second when he met fellow former Manchester City man Kolarov’s free-kick with a free header.
The momentum was against Chelsea, but Conte’s men roused themselves.
Pedro crossed and Hazard found himself in space to head into the bottom corner.
Roma might have claimed a deserved victory, but Dzeko headed wide at the near post.
Courtois, Azpilicueta, Christensen, Cahill, Zappacosta (Rudiger 77), Fabregas, Luiz (Pedro 57), Bakayoko, Alonso, Morata, Hazard (Willian 80).
Alisson, Bruno Peres, Fazio, Juan Jesus, Kolarov, Nainggolan, Gonalons, Strootman (Florenzi 83), Gerson (Lorenzo Pellegrini 73), Dzeko, Perotti (El Shaarawy 88).
Damir Skomina (Slovenia)