A win in this one therefore was vital but the players were frustrated for much of the first 45 minutes.
The Potters arrived seemingly determined to get a goalless draw, sitting behind the ball and showing no attacking ambition.
The 19-year-old drew two early stops from Asmir Begovic and was a constant menace on the wing.
Begovic then has to be alert to palm away a Leon Osman shot that had taken a wicked deflection off a Stoke defender.
Frustration began to grow as the game approached half time, but that tension was lifted in the final minute of the half.
It was the perfect time to score and the momentum gained continued after the break, with Everton doubling their advantage within four minutes.
Deulofeu was again at the heart of things, racing into the box before the ball fell to Seamus Coleman, who mis-hit the ball into the net with his left foot.
Despite being two goals down Stoke showed no ambition to try and get back into the game and were comfortably the worst side to visit Goodison this season.
It was no surprise therefore when Everton made it three on 58 minutes.
Bryan Oviedo, in for Leighton Baines, collected a short corner on the edge of the box, shifted it onto his right foot and fired home in off the right-hand post from 20 yards.
The Costa Rican put in an impressive shift (helped by Stoke’s ineptness, obviously) and set up the fourth, bounding down the left before teeing up Romelu Lukaku, who sidefooted home from close range.
Lukaku was immediately replaced by Nikica Jelavic and the afternoon would have been topped off by a fifth from the Croatian, but Begovic saved his low shot after being put clean through in the final minute.
After four minutes of injury time Stoke were put out of their misery and Everton claimed as win that sends them into the top four.
There are bigger challenges to come, not least in the next seven days, but these sorts of wins are just as important to keep the points tally ticking over.
Given Ross Barkley and Kevin Mirallas were only substitutes, Evertonians can look forward to Old Trafford and the Emirates with a rare sense of optimism.
Howard 7 Coleman 8 Oviedo 8 Distin 7 Jagielka 8 Deulofeu 9 Pienaar 7 (Mirallas 83, 6) Barry 8 McCarthy 7 (Stones 74, 6) Osman 7 Lukaku 8 (Jelavic 80, 6)