Michy Batshuayi gave Chelsea a 1-0 lead midway through the second half only for Jamal Lewis to force extra time with the final kick of the game.
But it was in extra time where there was plenty of controversy which started when Willian was booked for diving in the box, although pundits commentating on the game thought it should have been a penalty.
Then Pedro picked up two yellow cards, the first for a clear dive in the penalty area.
And then Alvaro Morata picked up a quick-fire double yellow again for an adjudged dive in the box and then overzealously protesting his innocence to the referee.
There was little time for Norwich to make the most of the advantage of having two extra men on the park with the game decided by a penalty shoot-out.
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Willy Caballero proved the hero of the night by saving the Canaries’ first penalty from Nelson Oliveira, and Chelsea failed to miss with Hazard scoring the decisive spot-kick.
And the Belgium international was happy that his side were able to get the job done in the end.
“Now we can focus on the league and get ready for the next game against Newcastle,” Hazard told BBC.
“It is the FA Cup it is not easy to win the game.
“We gave everything, at the end of the game we played nine vs 11 so we showed great character.
“We are happy but also sad that we lost two players.”