Fulham can breathe easy, at least for a little while. This victory, courtesy of goals from Darren Bent, Ashkan Dejagah and Steve Sidwell, ensures a trip to Sheffield United in the fourth round but perhaps more significantly allows the gloom threatening to engulf their Premier League campaign to be forgotten for a few days.
How Rene Meulensteen will be glad of it. Next up for Fulham is a trip to the Emirates before games against Swansea City, Southampton, Manchester United and Liverpool. For now, though, Meulensteen can reflect on an evening where his team played coherently and with spirit, even if the challenge posed by the opposition was weak.
This was not arguably a vintage display from Fulham, but it did not need to be. Norwich were extremely poor. This defeat made it no wins in eight attempts for Chris Hughton's side who, while one point above Fulham in the league table, looked bereft of confidence and lacking ideas.
It was an indicator of Meulensteen's and Hughton's priorities that there were a combined 15 changes from last weekend's league meetings. Fulham welcomed back Brede Hangeland after a three-month absence and the defender could play at Arsenal this weekend.
Meulensteen, who confirmed that the club had made a formal bid for the West Ham midfielder Ravel Morrison, said Hangeland's return was a major boost and he was also pleased with his fellow central defender Dan Burn, who made his home debut.
"It was a very big boost. Brede's done ever so well after surgery and trained very hard. He's been working hard to get back. It's great to see that he came through the 90 minutes very well," said Meulensteen.
"You need to put the results in the right perspective. We are doing a lot of things well. I think on Saturday [against Sunderland] we were dominating and controlling and should have had our noses in front. We set out today to make sure we came out with a clean sheet."
Norwich hit the crossbar in the third minute through a dipping shot from Robert Snodgrass but that was as good as it got for them. Fulham deserved their lead and all three goals were relatively straightforward.
Bent's finish was a simple strike from the centre of the penalty area after 17 minutes. Pajtim Kasami drove to the by-line and swept a low ball across goal that evaded Dejagah but found Bent in the right place to finish well.
Such was the paucity of Norwich's attacking threat that the away fans resorted to gallows humour when their side earned a corner with five minutes of the half remaining, cheering incessantly.
They were not laughing, though, when Fulham cleared the set piece and broke quickly down the right, Alexander Kacaniklic dragging the ball across goal for Dejagah to score from close range.
Sidwell completed the scoring with 22 minutes to play. A swift ball in was delivered from the left and Dejagah was almost on hand to divert it past Bunn once again, only for Javier Garrido to get the faintest of touches and divert it out to the opposing wing. Garrido's intervention, while impressive, only delayed the inevitable as Sascha Riether then hung up an inviting cross to the far post where Sidwell finished with an emphatic header.
Hughton said: "I will always take full responsibility myself. I'm the manager of this team, we win together and we lose together. The only way you alleviate pressure is by winning games."