The Belgian international netted twice in the first half to cancel out Sol Bamba's shock opener, before converting from the penalty spot 10 minutes from time to kill off the match.
Having trailed for the first time this season, the Blues showed good character to hold off Cardiff and added a late fourth through substitute Willian, lifting them above Liverpool at the top of the standings on goal difference.
Maurizio Sarri's only change from the 2-0 win over Bournemouth would prove a decisive one as Olivier Giroud - in for the benched Alvaro Morata - played a key part in both of Chelsea's first-half goals.
Neil Etheridge was equal to both efforts, producing a fine stop in particular to thwart Kovacic down low when the Croatian midfielder was played clean through.
Another chance would come and go for Pedro, who this time skied the ball unmarked from eight yards out, but Chelsea's pressure would eventually tell.
Hazard scored his first of the afternoon 37 minutes in when exchanging passes with Giroud and drilling the ball across Etheridge into the bottom corner, before tucking home a second via a deflection ahead the interval after being teed up by Giroud from closer range.
Chelsea were unable to get out of first gear for large parts of the second period, taking until the hour mark before registering their next shot through Pedro.
With just the one goal in it with a little over a quarter of the match to play, the Blues - who had nine regulars away on international duty last week - were unable to truly relax.
Instead, Hazard completed his hat-trick by sending Etheridge the wrong way from the penalty spot, after Bamba brought down Willian inside the box, and the Brazilian substitute rounded off the scoring with a long-range curler.
A Kepa save from Callum Paterson's header meant a comfortable conclusion to the match for Chelsea, and a third successive 4-1 defeat here for the Bluebirds in all competitions.
CHELSEA (4-3-3): Kepa; Azpilicueta, Rudiger, Luiz, Alonso; Jorginho, Kante, Kovacic (Barkley 50'); Pedro (Willian 68'), Giroud, Hazard (Zappacosta 84')
CARDIFF CITY (4-4-2): Etheridge; Manga, Bamba, Morrison, Bennett; Camarasa, Ralls, Arter (Richards 46'), Hoilett; Ward (Madine 77'), Reid (Paterson 71')