Defender Lamine Kone struck twice from close range after Patrick van Aanholt scored with a 25-yard free-kick that was misjudged by keeper Joel Robles.
Victory capped an impressive revival by Sam Allardyce's side, who have lost only two of their past 13 games.
Everton were dire, heaping more pressure on boss Roberto Martinez.
The Toffees have picked up only six points from their past 10 league matches and the Spaniard is likely to face an uncomfortable final game of the season in front of disgruntled crowd at Goodison Park.
In contrast, Allardyce will be able to take his side to Watford on Sunday unburdened by the relegation fears that have surrounded the club since his appointment in October.
Their victory leaves them four points ahead of both Newcastle and Norwich with just one game to play.