Midfielder Dusan Tadic’s 69th-minute goal earned the Saints a first league win at Manchester United for 27 years and lifted them above their top-four rivals in the table.
The high-flying south coast side – tipped for a relegation battle after selling £100m of stars in the summer – have now beaten United, Arsenal and Everton and drawn with leaders Chelsea.
“We know that we have a strong team and that was important in the end of December,” said Koeman.
“We got a point against Chelsea and the three points against Arsenal. That makes belief in the players. That is what you need if you are not a big club, belief in the players, not afraid of the name or the stadium. We learned that.
“I am not surprised we won today. We have organisation. If we keep the spirit and the organisation, we can keep in front in the table.”
Koeman’s triumph was all the sweeter as it came at the expense of United boss Louis van Gaal, having fallen out with his fellow Dutchman while at Ajax more than 10 years ago. Defeat was United’s first at Old Trafford since the first day of the season and left them with 37 points from 21 games – the same as David Moyes managed in his doomed term last year.
“You have been waiting for the moment to tell me I have the same points as David Moyes,” bristled Van Gaal, who said Southampton had been lucky despite his team failing to register a single shot on target.
“It is a bad result. We have to be better. We were the dominating team but you have to create more chances when you are the dominating team.
“We played a much better game than Southampton. They were lucky to score out of nothing. They came for a draw, I believe, and they go away with a victory. That is disappointing.”
Tadic calmly slotted through a goalmouth crowd after striker Graziano Pelle, another player signed to replace the close-season exodus, had struck United goalkeeper David de Gea’s right-hand post.