The loss of manager Mauricio Pochettino and a host of key players during the close season had prompted many to predict relegation for the south coast side.
However, 25 points from a possible 33 represents their best return at this stage of a Premier League campaign.
Only Jose Mourinho’s table-topping Chelsea lead Southampton by four points and, according to Koeman, that should be no reason for raised eyebrows.
“It’s still only 11 games played but, of course, nobody is surprised about the result of today,” the 51-year-old told Sky Sports.
“Everybody knows that Southampton has a very good, strong team and if we live game by game, we can be successful and (our) position in the table, we will see at the end of the league.”
Substitute Shane Long scored twice as the Saints beat a plucky Leicester side to maintain the pressure on Chelsea.
Southampton laboured before the Irishman’s introduction but Koeman said the team’s patience and trust in their quality saw them home.
“We were not good enough in the first half to score and we lacked a bit of quality in the final pass,” he said.
“But we spoke to each other at halftime to say ‘OK, we have to play out of a good organisation and we will create opportunities to score because we have good players’.
“Of course we had to be very patient today and that was the quality and that was the key for winning.”
Remarkably, despite selling defenders Luke Shaw, Dejan Lovren and Calum Chambers in the close season, Southampton’s lofty position in the table has been built upon a mean defence which has conceded five goals in 11 Premier League games.
That is five less than the Premier League’s next best defence, Swansea, who have played one game fewer.
“Yes (the run of clean sheets) is fantastic. Again today Leicester had only one big chance in the final (part) of the game,” Koeman added.
“Another clean sheet means a lot about our organisation and qualities in defence. It’s going well because I always say clean sheets make you a winning team. In front we have qualities, we have good players to make a difference and you can see that.”
(Reporting By Tom Hayward; Editing by Ian Chadband)