Victor Wanyama was the beneficiary of a mistake from Hull's third-choice keeper Eldin Jakupovic in the third minute, with his poor kick falling straight to the Kenyan who superbly guided a 40-yard effort into the unguarded goal.
That proved enough for Southampton to rack up a 10th win in their last 11 matches, and keep them second in the table.
Southampton's supporters greeted the final whistle at the KC Stadium with a chorus of 'we're going to win the league', and while manager Koeman is surprised by their superb start to the season, he feels their team spirit can help keep them at the top end of the top flight.
"Maybe, maybe [we can challenge for the title]," he said. "[We are] always positive. [If we] continue winning, continue up in the table then why not?
"Because we have a good team organisation. I think we showed that again.
"In the first half after the incredible [goal from] Victor, we were maybe a bit sloppy after that for 20 to 25 minutes and Hull dominated but didn't create any good chances, and didn't create any chances in the second half.
"The expectation is high. I like that because we like to win. We are football players and we like to win games. If you win games you will have good possibilities to maybe play in Europe, you never know.
"Everyone knows we have to work hard every day. The players are incredible in the way they act together, the spirit we showed today is important. We have to keep that."
Koeman also paid tribute to Wanyama for his goal and all-round contribution to Southampton's cause.
"He was fantastic," he added. "The goal was amazing, but also how he played today.
"He was strong, winning the battles, playing good on the ball, he had good positional play. It's fantastic to have those kind of players."
