Liverpool have started with six wins in the Premier League and a victory in the Champions League but they were knocked out by a superb solo goal from Hazard in the 85th minute.
Former Chelsea striker Daniel Sturridge, who had earlier missed an open goal, put Liverpool in front in the 58th minute with an acrobatic volley after Willy Caballero parried out Naby Keita's drive.
Emerson levelled for Chelsea in the 79th, slotting home after Simon Mignolet parried out a Ross Barkley header from a free kick whipped in by Hazard.
The goal was reviewed by VAR but ruled as fair although Liverpool manager Juergen Klopp felt it should have been disallowed for offside.
Sturridge then struck the bar with a fine effort from distance but it was Chelsea who struck the decisive blow.
Belgian forward Hazard, a 56th-minute substitute, won the game with a devastating finish after a powerful run in from the flank, weaving past two Liverpool defenders behind before firing into the far corner.
"It's a fantastic goal. Not only the beauty of it but for the time he scored it," Chelsea assistant manager Gianfranco Zola said.
"Not only is he one of the best in the Premier League - if not the best - but he is one of the best in Europe and the world.
"In my opinion we haven't seen the best of him yet. He's remarkable," he added.
The two teams face each other at Stamford Bridge on Saturday in the Premier League.
Elsewhere, West Ham United hammered fourth tier (League Two) Macclesfield Town 8-0 for their biggest win since 1983.
Grady Diangana and Robert Snodgrass both scored twice as West Ham ripped apart the Cheshire side who are bottom of League Two without a win this season.
In an all-Premier League tie, Tottenham Hotspur progressed to the last 16 by beating Watford 4-2 on penalties after the game, played in Milton Keynes, ended 2-2.
Isaac Success had given Watford the lead in the 46th minute but Spurs looked to have grabbed the win thanks to a Dele Alli penalty in the 82nd minute and Erik Lamela's goal four minutes later.
But Etienne Capoue's 89th-minute drive took the game to penalties, only for Capoue and Domingos Quina to fail to convert their spot-kicks and Alli wrapped up victory for Spurs.
Danny Welbeck scored twice as Arsenal beat Championship Brentford 3-1.
After the former Manchester United striker put the Gunners 2-0 up, Alan Judge brought Brentford back into the game but Alexandre Lacazette made sure of the win for Arsenal.
In an all-Championship match, Nottingham Forest beat Stoke City 3-2 despite having Diogo Goncalves sent off 13 minutes from time.
(Reporting by Simon Evans, editing by Ed Osmond)
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