Wayne Rooney marked his top-flight return to Everton after 13 years by scoring the winner against Stoke and his 199th goal in the Premier League.
The 31-year-old, who left the club for Manchester United in 2004, found space in the area to head a Dominic Calvert-Lewin cross past Stoke goalkeeper Jack Butland before the break.
It was the Toffees' best move of the match - one Rooney was twice involved in before adding the finishing touch.
Calvert-Lewin also went close for the home side as Stoke struggled to break Everton down on the opening day at Goodison Park.
Xherdan Shaqiri's long-range effort did force new Toffees goalkeeper Jordan Pickford into a fine save in stoppage-time, with the 23-year- old leaping across his line to tip wide.