Rooney, who was full of invention and energy throughout, rose well to meet a fine Dominic Calvert-Lewin cross with a looping header into the far corner on the stroke of half-time.
The former England captain rejoined his boyhood club last month after 13 years with Manchester United and he instantly looked at home.
"To play for this football club again is a huge deal for me and to score at Goodison in my first game back, it doesn't get much better. A great moment," said Rooney.
"It was a tough game and we were a bit fortunate to go in 1-0 up at halftime. I thought we were the better team in the second half and deserved to win," he added.
The 31-year-old, playing just behind debutant Spanish striker Sandro Ramirez, delivered some clever passes and fed the lively Calvert-Lewin with some perfectly weighted through balls.
"He was clever, comfortable on the ball. That is why we signed Wayne, expecting to show his team-mates all that experience, and it was really the case today," said Everton manager Ronald Koeman.
But while Everton dominated large parts of the game, they needed new goalkeeper, Jordan Pickford, to pull off a superb save in injury time to deny Xherdan Shaqiri an equaliser from a long-range drive.
Earlier Calvert-Lewin, the England Under-20 international, who again showed his great potential, could have made sure of the win in the 69th minute when he robbed Kurt Zouma on the edge of the area. But his fierce blast was kept out by Stoke keeper Jack Butland.
Stoke battled hard and were as well-organised and dogged as ever but lacked the creative touch in the final third.
"Overall I thought we did OK ... I didn't think there was too much between the two sides," said Stoke manager Mark Hughes.
"Their keeper made a great save at the end to save the result for them."
But Hughes said there was something inevitable about Rooney's role in the win.
"He was always going to be part of the story ... that is what we feared," he said.
(Reporting by Simon Evans,; Editing by Neville Dalton)
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