Another loss for Everton means that it is one win in 13 games as the pressure is at an all-time high on boss Rafa Benitez.
Two goals in quick succession for Norwich secured the three points for Dean Smith.
An own goal from Michael Keane and a first Premier League goal from Adam Idah gave Norwich the lead in the first half.
A goal from Richarlison on the hour mark gave Everton some hope but they could not build on it as Tim Krul had little to do.
After just 16 minutes Everton had fallen behind after a slow start to the game from both sides.
Norwich City were attacking down the right as Josh Sargent crossed the ball into a dangerous area and Keane stuck a desperate leg out inadvertently putting the ball past Jordan Pickford into his own net.
Two minutes later Everton lost the ball in the final third and Norwich in the transition split their defence open thanks to one pass from Brandon Williams and Idah was in due to a massive gap in the middle of the away sides defence.
Idah took a touch and scruffily finished as Pickford attempted to charge out to block the attempt but he couldn’t get near it.
Everton were in a state of shock as Norwich scored their first league goals in nine and a half hours of action.
For the rest of the first half Benitez’s side struggled to mount a serious attack together whilst they were in a state of panic every time they had to defend.
The second half started in the exact same fashion as The Canaries should have finished the game off within the first ten minutes.
Vitaliy Mykolenko played a blind back-pass straight to Idah who tried to get round Pickford instead of taking the shot on and the chance escaped him so therefore the game was still on.
On the hour mark soon after Idah wasted that massive opportunity substitute Richarlison dragged The Blues back into the game with a fantastic overhead kick after Tim Krul punched the ball away, but he did not get enough distance on his clearance.
Everton struggled to sustain attacks in the last half an hour as they were searching for an equaliser and Anthony Gordon had the best chance.
After some slow build-up play the ball got pulled back to Gordon whose shot was heroically blocked by defender Dimitrios Gianmoulis.
After that there was no clear-cut chances as Everton resorted to hopeless long balls and hopeful stray crosses.
In the seven minutes of stoppage time Norwich’s Milot Rashica had the best chance as he went through in goal only to clip the outside of the post.
It did not matter though as straight after that referee Andrew Madley blew his whistle as Dean Smith kept up his unbeaten record against Everton and got Norwich City’s first points in six games.
They now sit in 18th place one point off survival and only six points off Everton.
Rafa Benitez’s Everton will be very much looking over their shoulder as a relegation battle looks likely after one win in 13 league games.
The Spaniard walked straight down the tunnel once the game ended, which is not a surprise as if he went to applaud the away fans the hostility would have been something else.
Up next for Everton is Steven Gerrard’s Aston Villa and a reunion for Lucas Digne.
Norwich City XI: Krul, Aarons (Giannoulis 80), Hanley, Gibson, Williams (Byram 77), Lees-Melou, Soerensen, Sargent (Tzolis 87), Idah, Rashica, Pukki
Everton XI: Pickford, Coleman (Mina 54), Godfrey, Keane, Mykolenko, Gordon, Doucoure, Gomes, Gray, Rondon (Richarlison 54), Calvert-Lewin
