Graziano Pelle has capped the week where he scored on his Italy debut by being named Premier League player of the month too, in a double reward for Saints' impressive early season form.
New boss Koeman has shaken off widespread pre-season jitters to guide Southampton to third in the league, all after a summer clear-out with top stars following manager Mauricio Pochettino out of St Mary's.
The former Holland defender 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 Saturday's home Premier League clash with 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.
"Maybe Liverpool were surprised in the first match of the season, but after that, everybody knows that still Southampton has a very good team and can beat every team in the Premier League.
"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.
"And that's more pressure, that's more pressure for everybody."
Koeman brought his former Feyenoord striker Pelle to St Mary's in one of his early transfer moves on the south coast.
The 29-year-old has struck four times in seven league appearances, his acrobatic volley in the 2-1 win over QPR bound to contend for the goal of the season prize.
Pelle claimed the winner as Italy edged past Malta 1-0 in midweek, before now landing the player of the month gong.
Sunderland head coach Gus Poyet has hailed John O'Shea's leadership skills after watching him make his own way as a captain rather than adopting the Roy Keane approach.
The 33-year-old central defender signed a two-year contract extension on Thursday as the Black Cats tied up the immediate future of a man who has become central to their ongoing project.
Poyet was thrilled to have secured the services of a model professional, whose upbringing at Manchester United helped to hone the personality traits which have seen him establish himself as the management's representative on the pitch.
The Uruguayan said: "It's the whole thing, definitely the influence of Manchester United because he was there playing with top players, and I am sure he took things from different characters.
"He cannot imitate - he cannot be his previous captain, Roy Keane - but he can imitate other ones
It's just to learn and put what you are into the job.
"He does it naturally, I can tell you, there are no issues."
O'Shea, who features in Keane's updated autobiography The Second Half with the Republic of Ireland and Aston Villa assistant manager jokingly claiming he helped to bring Cristiano Ronaldo to Old Trafford by playing like "a clown" against the then Sporting Lisbon youngster in a pre-season friendly, capped a perfect few days by putting pen to paper on Wearside.
The contract announcement came less than 48 hours after he marked his 100th cap for Ireland by scoring a late equaliser against world champions Germany in Gelsenkirchen, illustrating the range of qualities Poyet values so highly.
Asked if O'Shea is the perfect role model for his younger players, the head coach said: "I'll tell you what, it's easy that answer because yes, it's true.
"He's always there, he's always on time, he is always fit - or most of the time - he looks after himself, he represents the club well, the players
That's why he is the captain."
O'Shea's midweek heroics will be a distant memory on Saturday when he leads the Black Cats out at Southampton looking for follow up a first league victory of the season at home to Stoke with a second at Southampton.
Steven Fletcher's double - his first goals since December 28 last year - helped to secure a 3-1 win over the Potters and end a miserable injury-plagued spell for the Scotland international.
Poyet will be without full-back Billy Jones, who suffered a muscle injury last week, and Santiago Vergini, who won his second cap for Argentina in midweek, is likely to deputise.
Source : PA
Source: PA