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.
The travelling fans made clear that they feel only the departure of Benitez – a controversial summer appointment given his six-year spell at rivals Liverpool – can reverse the slide.
Norwich, whose last Premier League goal was back at the end of November, are one point off 17th place and safety.