Wayne Rooney was the match winner on his return to his boyhood club with a well-timed header as the match ended 1-0.
While Koeman hailed the result, he admitted his side ‘didn’t play well’.
He told Sky Sports: “I’m happy with the points. Maybe every manager before the start of the season thinks that their team is ready and prepared but it was really a difficult game.
“Stoke made it really difficult and if you know that, and you keep the clean sheet and the points at home, it is a perfect result. It was a win and that’s what you like to have after the first match.”
He added: “I think it is a nice challenge for everybody, maybe it bought a few more nerves to the team as it was the first time for the points and playing at home.
“Maybe that was the reason we didn’t play well but I think the second half was much better. It was much better controlled and I think we had some good counter-attacks to score a second goal.”
On Rooney’s contribution, Koeman said: “We know he is one of the best players in his position and he showed today the player that we expected with a lot of cleverness on the pitch.
“He made good decisions on the ball and scored a great goal. It was a great cross from Dominic and a great header but we can ask for more than he showed today.”