Koeman slams Sheff United 'lack of respect'
Koeman's side travelled to Bramall Lane looking to secure a place in the League Cup semi-finals, but suffered a fifth successive defeat in all competitions.
Mark McNulty netted the game's only goal in the 63rd minute, pouncing on an abysmal mistake from Southampton goalkeeper Fraser Forster.
United assistant Chris Morgan was sent to the stands almost immediately after the goal as Koeman found himself in a verbal battle with the hosts' backroom team.
And in response to what Koeman felt was an absence of reverence, the former Barcelona defender refused to shake the hand of opposing manager Nigel Clough at full-time.
He explained: "I shake hands with people who have respect for me as a coach, who have respect for the fourth referee [official].
"I think the behaviour of the bench of Sheffield United, I never saw that. That was the reason why I didn't shake hands."
To make matters worse for Koeman, Southampton will be without suspended trio Morgan Schneiderlin, Victor Wanyama and Florin Gardos as they look to avoid a sixth consecutive defeat when hosting Everton on Saturday.
"It's difficult," acknowledged Koeman. "In my opinion that's one of the problems in the last few weeks – we have to change every time the team and with injuries with suspensions and, okay, suspensions is part of football and injuries is a part of football, but we have not a lot of midfielders available.
"We had to make choices. Schneiderlin played finally 90 minutes. Normally he wasn't allowed to play 90 minutes but we had to change and that was the problem and we knew that before the season.
"If we get these kind of problems in injuries and suspensions then it's difficult."