The only goal of the game came ten minutes after break, Michael Keane heading in from Lucas Digne's corner.
The result lifts the Toffees into the top 10, albeit momentarily with Burnley and Arsenal yet to feature in the latest round of games.
Story of the match
The first chance of the game came for the away side with Dominic Calvert-Lewin struggling to direct his header goalwards after a brilliant cross from the right by Bernard.
Norwich's first opportunity of the night came through Onel Hernandez who weaved his way towards the box before seeing his shot deflected out for a corner after great defensive work from Mason Holgate.
The game struggled to get going for much of the first half with both sides not looking like threatening either goalkeeper as the opening half went on.
Just before the break, Calvert-Lewin was involved, taking it on his chest before putting in a cross towards the back post, it was nearly met by Richarlison but the ball evaded the Brazilian.
The best chance for the hosts came just before the break as Lukas Rupp got hold of an effort from just outside the box but Jordan Pickford was equal to it and able to make a punch away.
It took the Toffees just ten minutes in the second half to take the lead, Michael Keane heading in from Lucas Digne's corner.
Calvert-Lewin nearly made it two moments later, shrugging off his marker before firing a shot goalwards which was turned over the top by Tim Krul.
Norwich tried everything and had a great chance from a free-kick. Andre Gomes nearly sent it into his own net but it went just wide of the post.
Everton continued to dominate and saw the game out well to secure a crucial three points.