Chelsea, who had lost two of their previous three Premier League games, were dominated by high-flying City until the 44th minute when Luiz beat the visitors' high-pressure tactics with a pinpoint 50-metre pass over the midfield to Pedro.
The ball fell to Hazard who rolled a pass into the path of Kante to thump it past Ederson. The Brazilian goal-keeper had not had to make a save until that point.
It was Kante's first goal since the opening game of the season and could ease some of the criticism of the France midfielder's suitability for his new, more attacking position than his trade-mark anchor role which has gone to Jorginho.
City had cut Chelsea open on several occasions but, with striker Sergio Aguero sidelined due to injury, they failed to convert their chances in the opening period.
The goal gave out-of-form Chelsea -- also playing without either of their established centre-forwards - a jolt of confidence for the second half.
With the home fans increasingly sensing an upset win, Luiz headed home Chelsea's second in the 78th minute from a corner by Hazard, the ball looping in off his rain-soaked mop of hair and sealing City's first league defeat since April.
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
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.
(Reporting by Clare Lovell, editing by Ed Osmond)
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