Roberto Martinez's side recorded its biggest win of the season as the Potters' poor run of form continues with just one win in 10 Premier League games.
The Barcelona loanee, who was handed his first Premier League start after a bright display as a substitute against Liverpool last week, was the shining light in a vibrant attacking display that lifts Roberto Martinez's side up to fourth in the table.
Everton dominated from start to finish and Deulofeu opened the scoring just before the break by finishing off a superb team move, before second-half goals from Seamus Coleman, Bryan Oviedo and Romelu Lukaku ensured the club made it 16 home top-flight games without defeat.
Martinez made three changes from the thrilling 3-3 draw against Liverpool last week, with Leon Osman, Oviedo and Deulofeu coming in for Kevin Mirallas, the injured Leighton Baines and Ross Barkley.
Mark Hughes, meanwhile, made two alterations from Stoke's 2-0 win over Sunderland, as Marc Muniesa and Oussama Assaidi replaced Robert Huth and Marko Arnautovic.
Everton unsurprisingly started the game on the front foot, with the impressive Deulofeu proving a thorn in Stoke's side.
The 19-year-old had the first effort on goal when he forced Asmir Begovic into a smart save from his 20-yard free kick after just three minutes.
Begovic was once again called into action in the 26th minute as he made a superb reflex save from Osman's deflected shot following another sustained period of Everton pressure.
Martinez's men continued to control proceedings and finally broke the deadlock on the stroke of halftime when Deulofeu finished off a brilliant move from close range.
The Spaniard played slick one-twos with Steven Pienaar and Gareth Barry in the Stoke box before rifling past the helpless Begovic from six yards.
The home side picked up where it left off after the interval and had to wait just minutes to extend the lead through Coleman.
Deulofeu was once again the architect, as his elusive run down the left led to Barry not controlling the winger's cross, only for the ball to fall fortuitously to Coleman who scuffed his left-footed shot past Begovic.
Oviedo made it 3-0 shortly before the hour, as he received a short corner from Deulofeu before cutting inside onto his right foot and arrowing his first Premier League goal into the bottom corner from 25 yards.
Tim Howard was forced into his first save of the game from Peter Crouch soon after, but Lukaku had the final say when he rounded off another incisive move by slotting home Oviedo's cross from six yards 11 minutes from time.