Sol Bamba shocked the home crowd by opening the scoring after 16 minutes, but a Hazard double made it 2-1 at the break. He then wrapped up his hat-trick from the penalty spot in the 80th minute after Bamba felled substitute Willian in the box.
The Brazilian wrapped up the scoring and the points with a thunderbolt seven minutes from time to put Chelsea top ahead of Liverpool, who have also won five from five, on goal difference.
Eden Hazard can Carry Chelsea to the Title
He spent most of the summer linked with Real Madrid, but Eden Hazard hasn't let the speculation disturb his form in a Chelsea shirt. The Belgium international is playing well enough to carry the Blues to another league title.
His two goals in the first half continued a remarkable recent run:
Hazard is the player who provides Chelsea with inspiration. His ability to conjure magic in an instant can turn games around for the Blues.
After Bamba's goal and Cardiff's direct approach had left Chelsea players and fans frustrated, Hazard's neat finish for his first lifted the mood inside the stadium:
Once Hazard sets the pulses racing, momentum usually swings Chelsea's way. Along with his natural ability, Hazard is good enough to haul a solid squad firmly into the title-winning picture.
Olivier Giroud is Better for 'Sarriball' than Alvaro Morata
Giroud has to continue starting ahead of Alvaro Morata if Chelsea manager Maurizio Sarri's style of play is going to be successful. The former Arsenal target man is ideal for the intricate passing combinations Sarri has brought to the Bridge.
The Frenchman is excellent with his back to goal and links well with others. Despite his bullish frame and ability to compete in the air, Giroud is more flair player than old-fashioned battering ram.
The Frenchman's game is defined by clever touches and deft flicks to release runners around him. Few centre-forwards can put together neat one-twos as well as Giroud.
It's how things worked for Hazard's opener:
The pattern continued to help Chelsea seize the initiative:
Not only does Giroud's game help him play a bigger role than Morata, it also makes those around him better. When Giroud can make forwards as capable as Hazard, Pedro and Willian better, the rest of the league should worry.
Yet it's probably Morata who ought to worry more, since he shouldn't get his place in the starting XI back.
David Luiz will Continue to Undermine Chelsea's Defence
Chelsea have the firepower up front to win the title on Sarri's watch, but the Italian's decision to restore David Luiz to the heart of defence could ultimately cost the Blues.
Luiz rarely looked comfortable right from the off against Cardiff:
Such struggles have become worryingly familiar for Chelsea fans during the Brazil international's two stints at the club. It's why questions have to be asked about Sarri's decision to start Luiz ahead of Gary Cahill and Andreas Christensen.
There is no doubt Luiz has the technique in possession to suit Sarri's passing game. Yet he remains vulnerable enough in the air and against pace to remain the weak link in a back four perhaps not strong enough to outlast the main contenders in the title race.
Chelsea begin their UEFA Europa League campaign in Greece against PAOK on Thursday. Meanwhile, Cardiff host champions Manchester City in the Premier League next Saturday.