Manager Roberto Martinez said it went some way to alleviating the disappointment Deulofeu felt after the Merseyside derby last week, when he missed a couple of chances after coming off the bench in a 3-3 draw with Liverpool.
"I thought Gerard was ready (to start against Stoke)," Martinez said on the club's website (www.evertonfc.com).
"He was fairly disappointed with last week. From my point of view - and I told him - I thought he was outstanding when he came on against Liverpool, but he had two good chances and he felt he should have taken them.
"I felt in the week he was left with a negative feeling, which he didn't deserve. That's why I felt it was right for him to start and push that feeling away - and he was desperate to do that.
"I thought his goal was sensational - the moment of the game. The way he works for the team and the way he understands his role in the side was very, very impressive, so we are very pleased.
"Now he can relax over the weekend with a good feeling."
The win took Everton up to fourth place, although the teams around them in the table - Chelsea, in third, Manchester City (sixth), Southampton (seventh) and Manchester United (eighth) all play on Sunday.
"We are very, very pleased," Martinez said. "I was a little bit worried whether we'd be able to maintain the tempo and intensity from the previous game.
"After the derby, having another game at home can be a little bit difficult, and that was a little bit of a worry.
"But the way we controlled the game was very pleasing, as was the way we coped with the strength of this Stoke side.
"Our movement and the way we kept hold of the ball was very pleasing; certain players with real fresh legs like Gerard Deulofeu made a big impact."
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