Antonio Conte’s team looked on course for a comfortable victory at Stamford Bridge thanks to first-half goals from David Luiz and Hazard at Stamford Bridge.
But Roma’s Aleksandar Kolarov struck before half-time to spark a stirring fightback from the Serie A side.
Former Manchester City forward Edin Dzeko bagged a brace after the interval to give Roma the lead, only for Hazard to spare Chelsea’s blushes with a late leveller.
Chelsea remain two points clear of Roma at the top of Group C, but this was the latest in a growing list of alarming displays from the Premier League champions.
Blues boss Antonio Conte reportedly held a lengthy inquest with his squad in the aftermath of Saturday’s shock 2-1 loss at Crystal Palace and this erratic performance — which extended their winless run to three matches — may well spark another dressing down from the Italian.
“It was a difficult game against a good team. We suffered in possession and it wasn’t easy in the first half, despite scoring two goals,” Conte said.
“The change of the system wasn’t good, but when you are in a difficult situation sometimes you have to improvise.”
After being penned into their own half early on, Chelsea conjured an electric counter, but Alvaro Morata, showing signs of rust after his injury lay-off, shot tamely at Roma goalkeeper Alisson.
Roma were undaunted and Diego Perotti galloped forward for a stinging strike that whistled over.
Brazil defender Luiz had been missing in action during that Roma raid, but the part-time midfielder recovered his bearings to put Chelsea ahead in the 11th minute.
Stealing possession inside the Roma half, Luiz initially tried a defence-splitting pass, but when that was blocked, he seized on the rebound and curled a fine first-time finish past Alisson from 20 yards.
Roma midfielder Radja Nainggolan wasted a golden chance to equalise when he shot straight at Thibaut Courtois.
That miss seemed fatal for Roma when Hazard scored Chelsea’s second in the 37th minute.
It was Hazard who started the attack, pouncing in midfield and picking out Morata.
While Morata sized up his options, Hazard was sprinting into the Roma area, where he met his team-mate’s deft cross with a cool close-range finish.
– Warning signs –
Hazard’s first club goal of the season should have put Chelsea on cruise control, but they remained far from convincing and Roma were back in the game three minutes later.
Kolarov took possession wide on the left edge of the Chelsea area, surged past Cesar Azpilicueta and blasted a fierce strike that deflected in off Andreas Christensen.
Roma had won only once in their 15 games in England, but the way they were bossing possession suggested a memorable escape wasn’t out of the question.
The warning signs were there when Courtois was forced to stick out a boot to stop Dzeko converting Kolarov’s cross early in the second half.
Conte responded to Chelsea’s struggles by sending on Pedro for Luiz and moving Cesc Fabregas into a more defensive midfield role.
But the switch failed to stem the tide and Roma equalised thanks to Dzeko’s moment of magic in the 64th minute.
Federico Fazio spotted Dzeko’s run and lofted a perfectly-weighted high pass that dipped into the Bosnian’s path.
Dzeko contorted his body in mid-air to meet the ball with a sublime volley that flashed past Courtois.
Conte’s side were stunned, but Dzeko wasn’t finished yet and he put Roma ahead in the 70th minute.
Kolarov’s inswinging free-kick caught Chelsea flat-footed and Dzeko headed home from close-range.
But, just as Chelsea looked sunk, Hazard bailed them out, ghosting into the Roma area to equalise with a glancing header from Pedro’s 75th-minute cross.