Koeman clearly no longer considers returning star Wayne Rooney as an out and out striker and, rather, the Dutchman is looking for a collective effort to replace Lukaku's prodigious output.
The 41-cap global played just 13 times for his club past year as he had an injury-interrupted campaign, while his appearances for his country last October led to a war of words between Everton and Ireland team management.
Everton had earlier struggled for fluidity as Ronald Koeman fielded five new signings from the outset, with Jordan Pickford comfortable in claiming a clean sheet. Setting the bar too low could make fans, players and club owners question if they have the right manager for the job.
"Maybe a little bit more nervous than normally they are and also the way Stoke play, defending-wise they make it hard".
The opening day of a new football season is always special, but this one is particularly significant for me. Rooney will bring experience to the side and although he is not the player he once was, he will still be able to contribute greatly to the team.
The match was heading into the break goalless when the home side produced their best move of the game.
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.
"I know that expectations are higher than last season", Koeman told the Mirror.
Stoke are embarking on their 10th consecutive Premier League season.
A revitalised Wayne Rooney is relishing a new lease of life at Goodison Park after scoring the victor in Everton's 1-0 victory over Stoke City in their opening Premier League game of the season. "If you look at my team this weekend, and the bench, you can not compare it to the big six".