Chelsea manager Maurizio Sarri acknowledged his side had been lucky not to concede in the first half before improving after the break when Hazard, playing in the number nine role, helped Chelsea take the game to City.
Chelsea inflicted the first Premier League defeat of the season on champions Manchester City after goals from N'Golo Kante and David Luiz earned the Londoners an unlikely 2-0 victory on Saturday.
He urged Chelsea, who lost 2-1 to mid-table Wolverhampton Wanderers on Wednesday, to show the same intensity against less illustrious opponents.
"We will need to play with this level of determination and the right level of aggression in the other matches, otherwise it is useless to win a match like this," Sarri said.
Pep Guardiola's side had arrived at Stamford Bridge having seen their closest title rivals Liverpool leapfrog them into top spot with a thumping 4-0 win at Bournemouth.
City now sit second, one point behind Liverpool, but Guardiola was not too worried.
"The way we played today was incredible. I am so glad," he said. "We monopolised the ball. They beat us in one counter-attack and one set-piece."
Chelsea climbed to third on goal difference with Tottenham Hotspur set to play Leicester City later on Saturday.