Koeman told BBC Sport: "It was a difficult game. If you don't kill the game then it is always difficult until the last second.
"I think we played a very good first half. It was more difficult in the second half. Stoke changed their system and we didn't control like we had before. But we still had eight opportunities to score.
"You can't win every week so easy. Sometimes it is more difficult because Stoke were very strong. But it is a great win."
Southampton's next game is against the same opposition as today, the South Coast side taking on Stoke City in the League Cup at the Britannia Stadium on Wednesday.