Koeman has warned against complacency, admitting top-flight rivals are now "awake" to his side's threat.
"Maybe everybody was a little bit afraid, didn't expect the way we would play in the beginning," said Koeman, ahead of today's home game against Sunderland.
"But all the changes in the team, we had to bring in a lot of new players and then the team was working well. Maybe the people were surprised about that and maybe that's the reason that I've won this prize.
"The rest of the teams are now not surprised about our way of playing, they are not surprised about our qualities. Maybe they are more prepared to play against us than in the beginning.
"So I told my players we must start again this week - all our opponents are awake now and it's not easy. It's not 'oh, we'll see what happens' any more - the people expect now, a good Southampton team on Saturday.
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