The final six minutes were a complete nightmare for the Toffees, who conceded twice before the final whistle sounded.
Kei Kamara scored in the 84th minute to equalize for the host Canaries before Grant Holt scored on the final kick of the match.
Everton's lone goal was scored in the 39th minute by way of a Leon Osman header.
The win moved Norwich up to 12th place in the table while Everton stayed in sixth place but fell behind Arsenal for the fifth spot; they now trail the Gunners by five points.
Here are five lessons that were learned at Carrow Road on Saturday.