Neil Warnock's side remain winless this season and it's no surprise after their performance against the Canaries.
The Welsh club, recently promoted to the Premier League, were down 3-0 with 21 minutes remaining after Max Aarons slotted home for the visitors.
There was some reprieve for Cardiff, with Bruno Ecuele Manga bagging their first goal of the season after 77 minutes.
And speaking after the game, Norwich manager Daniel Farke was pleased with how his side controlled the game.
He said: "We were better in possession. It's so important that you take your well-deserved credit and today we were able to score.
"The result could have easily been a bit higher but I'm pleased with the 3-1 and pleased we're in the next round.
"We got the feeling we were pretty sharp and we wanted to go further on into the next round."