The striker headed his boyhood club in front in first-half stoppage time following a neat team move, which ended with Dominic Calvert-Lewin swinging a cross into the box and the veteran nodding it into the back of the net from close range.
Ronald Koeman's side started sluggishly and Stoke caused them a number of problems in the first half through Mame Biram Diouf and Xherdan Shaqiri, but clear-cut chances were in short supply at both ends of Goodison Park.
Rooney's header, which he placed beyond Jack Butland moments before half time, was the first effort of any note during the opening 45 minutes.
Stoke created their fair share of chances after the break, with new signing Darren Fletcher coming close just before the hour mark when his low drive from 20 yards flew inches wide.
Everton had the chance to kill Stoke off in the 69th minute when Rooney's measured pass sent Calvert-Lewin through on goal, but Butland proved equal to his shot, turning it behind for a corner.
Another Toffees debutant, goalkeeper Jordan Pickford, also proved instrumental in the victory, producing an impressive save in stoppage time to keep out a goal-bound strike from Shaqiri at full stretch.
The result marks Everton's first Premier League opening-day victory since the 2012-13 season, when they beat Manchester United by the same scoreline at Goodison.