It was a strike written in the script for the returning Toffees after 13 years away and, speaking to the Liverpool Echo, Koeman explained how Rooney's influence permeated a tough encounter with the Potters.
"We know we brought in a really good goalkeeper", Koeman told Sky Sports.
But in his first Premier League outing Rooney not only scored the victor with a beautifully executed header from a Dominic Calvert-Lewin cross, but he also exuded quality and composure throughout.
Rooney's goal on Saturday was assisted by a cross from Dominic Calvert-Lewin, who was 7 years old the last time Rooney scored for Everton. "A great moment", said Rooney.
England's all-time leading goalscorer struck with a thumping header on the stroke of half-time and produced a fine second-half performance to ease himself back into the old routine in royal blue.
He said: "There was one player in an Everton shirt today who was from the first second to the last comfortable on the pitch". "He made every decision the right one and he scored a great goal from a great cross from Dominic".
"He was clever, comfortable on the ball".
"That's why we signed Wayne, expecting to show his teammates all that experience and it was really important".
It was Stoke who went close to scoring first, however, when Darren Fletcher, making his own debut following a summer move from West Bromwich Albion, drilled a shot narrowly wide from the edge of the penalty area.
Calvert-Lewin almost doubled Everton's lead when he robbed Zouma and cut into the penalty area for a drive that Butland pushed away. "I didn't think there was too much between the two sides", said Stoke manager Mark Hughes.
The introduction of Peter Crouch for the anonymous Saido Berahino gave Stoke a menace they had previously lacked and it needed several last-ditch interventions by Everton to preserve their clean sheet. "I thought we were the better team in the second half and deserved to win", he added.
"He was always going to be part of the story today, that was our worry and how it panned out".