But Roma were never out of the contest and Aleksandar Kolarov’s goal five minutes prior to the break gave them hope of a second-half reprisal that came true when Dzeko scored a stunning volley and a header to give them the lead.
The contest swung from end to end throughout and even at 3-3 there were numerous chances for both sides to find a winner, but a stalemate seemed fitting after Hazard and Dzeko matched each other goal for goal and inspired their sides in equal measure.
Returning from injury, Morata fired straight into the arms of goalkeeper Alisson when Hazard played him through on goal in the fifth minute, but the visitors refused to sit back and Diego Perotti went on a mazy dribble through Chelsea’s defence three minutes later.
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Errors began to creep into Roma’s defensive play and after his ball into the box for Morata was blocked by Juan Jesus, David Luiz ran onto the rebound and stroked it beyond Alisson into the bottom right corner of the net from 20 yards.
Just as Roma began to make gains in midfield, Chelsea broke at the other end and Morata’s deflected pass looped over to the far post where Hazard was on hand to tuck the ball into the net from close range.
But three minutes later Kolarov, who picked up the ball on the left wing, rounded Cesar Azpilicueta and rattled in a shot that hit the top of the net via a slight touch by Andreas Christensen.
In the second half, Dzeko came within a whisker of turning a Kolarov cross into the net as it flashed past his toes inside the six-yard box
Perotti then curled a shot into Thibaut Courtois’ hands with half an hour left to play, before a moment of brilliance changed the complexion of the game.
Federico Fazio jogged over the half-way line and played a long, chipped pass forward and Dzeko watched it all the way into the box before meeting it with his boot laces and volleying an unstoppable shot over the head of Courtois.
Chelsea had barely reacted to the blow when Dzeko scored again six minutes later, applying a glancing header to Kolarov’s free-kick to take it past Courtois and into the far corner of the net from six yards out.
If Roma head coach Eusebio Di Francesco was delighted with that, he will have been equally appalled by the simple nature of Hazard’s equaliser.
The Belgian forward was left unmarked when second-half substitute Pedro sent over a routine cross, and Hazard’s header found the bottom left corner of the net with accuracy, prompting a scramble for chances in the last 15 minutes but neither side could force a winner.