Southampton sit second after nine games, four points behind leaders Chelsea.
Graziano Pelle, a Koeman signing, has netted six times in the league and added another two to his personal tally on Wednesday as Southampton progressed to the League Cup quarter-finals at the expense of Stoke City.
Southampton's form has surprised some people, given they lost the likes of Dejan Lovren, Adam Lallana and Rickie Lambert before the season began, but Koeman believes his side can continue to fight at the top of the table.
"Maybe a lot of people are very surprised because of the start of the season," he told a news conference on Thursday. "I had the belief in the players and team spirit.
"Maybe not to be second but we have good players, we like to play football, everybody enjoys how we play, that's very important.
"We have had a good start to the season, it has given everybody the right spirit, that's the best quality of this team.
"We know we can keep doing this, we're not worried, there are not a lot of the other teams are better than Southampton."
Koeman's side travel to Hull City this weekend with the Dutchman wary of a side who have collected impressive draws against Arsenal and Liverpool in recent weeks.
"They had two good results in difficult away games," he added
"We know it is a tough one, there are no easy games, so have to prepare ourselves well for this Saturday."
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