Daniel Sturridge had made amends for missing an earlier sitter by edging the Reds in front on the hour mark, only for the visitors to hit back in style.
Having initially been shut out of the game by the opposition, second-half substitute Hazard sparked into life and played a direct part in both late goals for his side.
The Belgian's free kick led to Emerson Palmieri levelling up against the run of play, and the in-form forward then took matters into his own hands with an individual goal of the highest quality to win the tie five minutes from time.
With one eye on their upcoming Premier League battle, the teams showed eight changes apiece from their most recent league outings, with full Liverpool debutant Fabinho one of eight players on the field making their first start of the season.
Chelsea started by far the better of the sides and bossed possession throughout the first half, but they could not take advantage of their early dominance.
The visitors were coming out on top in the midfield battle, largely thanks to the work of Cesc Fabregas to pull the strings from deep.
A searching pass from the Spaniard found the run of compatriot Alvaro Morata a quarter of the way through, but the out-of-form striker could not lift the ball over Mignolet from the one-on-one position.
Morata was also unable to squeeze in the rebound from a tight position, and he was thwarted by Mignolet - in for his first Liverpool appearance since January - from the game's next big opportunity.
Liverpool had just a weak Naby Keita attempt to show for their efforts up until that point, but the momentum soon swung their way and they would have felt just as aggrieved as their opponents not to have taken a lead into the break.
Mane peeled away from his marker and looked for the bottom corner with his header, but back-up stopper Caballero watched it all the way and ensured that the sides went into the break all square.
Liverpool pushed on from their strong end to the first half and should have been in front within seconds of the restart, only for Sturridge to slot the ball wide of an empty net once played clean through by a poor Christensen back-pass.
Caballero did well to force Sturridge slightly wide of the target, but that was a let-off for the Blues and they learned from it, as Morata headed narrowly over the bar from Willian's outside-of-the-boot cross moments later.
Some poor play from Ross Barkley led to the Reds' next opening, though, which Mane failed to make the most of as his shot hit the foot of Caballero before deflecting wide, and from the corner the ball hit a Chelsea player and landed on the roof of the net.
Maurizio Sarri turned to Hazard as the hour mark approached, but the change did not initially have the desired effect as Liverpool edged in front 60 seconds later.
Sturridge made no mistake on this occasion, sending the ball flying into an unguarded net from 12 yards with a scissor-kick after Caballero parried Keita's stinger into his path.
Liverpool held fellow unbeaten top-flight side Chelsea at arm's length for the next 20 minutes, before the visitors struck a cruel blow through a right-sided Hazard delivery.
From a free kick the hosts never felt should have been awarded, Barkley glanced the ball on target and Emerson was quickest to react to the loose ball, prodding it over the line from close range.
In a gripping tie, Sturridge nearly regained the lead for Liverpool when his effort lobbed Caballero but came back off the frame of the goal.
The comeback was then complete six minutes later thanks to Hazard's piece of brilliance, slaloming through a couple of opposition players and unleashing a fierce shot into the bottom corner from 15 yards to end Liverpool's cup run at the first hurdle for the second year running.
LIVERPOOL (4-3-3): Mignolet; Clyne, Lovren, Matip, Moreno; Fabinho (Salah 87'), Milner (Henderson 60'), Keita; Shaqiri, Sturridge, Mane (Firmino 71')
CHELSEA (4-3-3): Caballero; Azpilicueta, Christensen (Luiz 73'), Cahill, Emerson; Fabregas, Kovacic (Kante 64'), Barkley; Moses, Morata, Willian (Hazard 56')