Belgium forward Hazard coolly slotted home Chelsea's final penalty to earn an unassailable 5-3 lead after Willy Caballero had saved from Canaries striker Nelson Oliveira to finally end the Championship side's resistance.
Chelsea were clinical from the spot after a frustrating third-round replay that saw them miss a host of chances, have three penalty appeals turned down and Pedro and Alvaro Morata sent off.
Michy Batshuayi had given Chelsea the lead after 55 minutes with the Blues' first goal since Marcos Alonso's 84th-minute strike against Arsenal on January 3.
But Jamal Lewis levelled things up in the final minute of stoppage time, before chaos ensued in the extra period.
Referee Graham Scott and his VAR denied Willian and Morata penalties in the added 30 minutes – the latter joining Pedro in being shown red.
The nine men saw out the few remaining additional minutes and made no errors from 12 yards to set up a fourth-round meeting with Newcastle United.
HAZARD SCORES! THE BLUES GO THROUGH!— Chelsea FC (@ChelseaFC) January 17, 2018
Problems on the London Underground meant kick-off was delayed by 15 minutes but the hosts did not seem affected as Danny Drinkwater and Willian both fired wide inside the opening 10 minutes, while Cesar Azpilicueta was unable to turn in Tiemoue Bakayoko's flick from a corner.
Norwich initially struggled to make any kind of impact in the final third and the Championship side were grateful to Angus Gunn for keeping the game level, the England Under-21 international goalkeeper brilliantly tipping Drinkwater's long-range effort onto the bar after 25 minutes.
Despite Chelsea's dominance their recent problems in front of goal showed little sign of stopping, while at the other end Oliveira fired two warning signs towards Caballero's goal as Norwich finished the half strongly – the first clipping the bar on its way over after a mistake from Batshuayi.
The Belgian striker made up for that error 10 minutes after the restart with Chelsea's first goal in 331 minutes, Batshuayi beating Gunn from close-range as he met Kenedy's low cross.
Any thoughts that the goal would open the floodgates were quickly dispelled as Josh Murphy and James Maddison almost profited from Chelsea mistakes.
The post and arm of Caballero saved them, though, and the hosts should have been out of sight late on, but substitute Morata twice missed golden opportunities.
And Norwich made him pay as Lewis headed home his first senior goal, the teenager meeting Timm Klose's cross in the last seconds of normal time.
Chelsea thought they had a penalty early in the first half of extra time as Willian went down under Klose's challenge, but referee Graham Scott and his VAR ruled it was not a spot-kick – the Brazilian booked for simulation.
Their frustrations grew further as Gunn made two saves in quick succession to deny Willian and Morata – the Manchester City loanee equal to both low efforts.
Morata was again wasteful in the second period, before Pedro – who was booked in the first half for diving – was sent off for a rash challenge on Wes Hoolahan.
He was not the only one to see red either with Morata shown two yellows – the first for a dive, Chelsea's third for simulation in the match – and the second in the immediate aftermath for his reaction towards the official.