The winger's sixth goal in seven appearances ended Liverpool's ideal start to the season with a 2-1 victory on Wednesday (Thursday NZT) at Anfield.
It was the first of two meetings between the clubs this week, with a Premier League clash at Stamford Bridge to come on Saturday. "Not only the beauty of it but for the time he scored it", Chelsea assistant manager Gianfranco Zola said.
They made a decent fist of that against Chelsea before Hazard's wonder goal sealed the win for the visitors and closed off another route to silverware for Liverpool.
Terry was delighted with the goal scored by his former Chelsea team-mate and reacted to the strike on Instagram.
Manchester City got the defence of the trophy off to a good start, romping to victory away at League One Oxford.
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While Terry used words to hail Hazard's mesmerising skill, Didier Drogba was reduced to emojis, while Sergio Ramos, whose Real Madrid side covet the Chelsea forward, also showed his appreciation with a like.
"We knew how Chelsea wanted to play but we gave them too much space and weren't compact enough", Liverpool manager Jurgen Klopp said.
"You have two players that are clearly offside and they block my players".
He told Sky Sports after his Man of the Match display: "This result will give us more confidence because we drew with West Ham four days ago". We have to take it.
Juergen Klopp's Liverpool have won all six league games in while Chelsea's winning streak was ended by a goalless draw at West Ham on Sunday.
Klopp confirmed goalkeeper Simon Mignolet will make his first appearance of the season in Wednesday's cup tie as Alisson Becker rests.