An own goal by Everton’s Keane and Idah’s strike was enough to secure the hosts a crucial three points.
Everton managed to improve in the second half, and Richarlison closed the gap with a spectacular overhead kick, but it was only a consolation goal for the Toffees.
Norwich now sits in 18th place in the Premier League table, one point behind Watford, who drew 1-1 at Newcastle United.
Watford may well have two games in hand, but Smith’s side is still in with a chance of survival in the Premier League as Everton remain in 15th place.
They managed nine shots in the box, but were unable to break down Norwich’s tenacious defence.
Norwich will be looking forward to their crucial match at Watford, while Everton are desperate for a win in their next game.
The pressure on Benitez is now becoming too much for him to bear.