David Luiz provided a rare assist from open play to allow Kante an opportunity that he pounced upon before scoring a goal that trebles his previous best tally in a Premier League season.
Palace went into the game seeking to extend a three-match unbeaten streak that included a famous win over Manchester City before Christmas, but they struggled to prise the ball away from a Chelsea side determined to avoid a repeat of their Selhurst Park defeat last season.
Antonio Rudiger headed tamely into the arms of Palace goalkeeper Vicente Guaita before Kante sent a more forceful shot wide as Chelsea enjoyed the better of the opening exchanges.
Olivier Giroud had the ball in the net after 27 minutes, lashing home from a tight angle only to see an offside flag raised against him.
The best attempt of the first half came from Willian, whose wickedly curling free-kick from 25 yards was saved by Guaita at full stretch just moments before the goalkeeper denied the Brazilian again when he tried a low drive.
The deadlock was broken after 51 minutes when David Luiz's exquisite chipped ball into the box was controlled instinctively on the chest by the on-rushing Kante, who then applied a low, left-footed finish that sneaked past Guaita despite the keeper getting a hand to it.
Guaita made another save from Willian, denying the winger when Hazard played him in for a shot, before another unfortunate moment for Giroud, who twisted his ankle in the process of scoring another goal that was ruled out by the assistant referee's flag.
The Frenchman limped off and was replaced by Alvaro Morata as Chelsea saw out the closing stages of a comfortable victory without any real threat of a Palace comeback.
Leicester was just a blip
Chelsea's 1-0 home defeat to Leicester City left Blues fans concerned about inconsistency under Maurizio Sarri, but this was a consummate away performance from a side with a strong gameplan and plenty of confidence.
Kante has it all
Not content with being the driving force in the engine room of Chelsea's midfield, Kante is making a habit of spotting gaps in opposition defences and proved he is adapting to his more advanced role this season with a worthy match-winning goal.
