Norwich rediscovered their scoring touch to end a six-game losing run, bolster their survival hopes and pile the pressure on Everton manager Rafael Benitez.
The Canaries took the lead when visiting defender Michael Keane sliced a clearance past his own goalkeeper.
Less than two minutes later, Adam Idah slid home a second to spark delirium around Carrow Road.
Everton substitute Richarlison gave the visitors hope with an acrobatic overhead kick after the break, but the gloom descended on the away end once more at the final whistle.
After a promising start to the season, Everton have lost nine of their past 12 Premier League games, slipping from fourth to 15th in the table in the process.