Graziano Pellè’s first-half double put Koeman’s men in control, before Marko Arnautovic pulled one back just after half-time for the Potters.
It set up a pulsating second half, but Saints – despite seeing Sadio Mané sent off – kept the hosts at bay to move above them in the Barclays Premier League table.
And Koeman was delighted with the win. “It’s a huge result,” he told youtube.com/southamptonfc.
“We spoke to the players before the game. This one and also the next one against Liverpool are very important, because that will mark the rest of the season.
“Now with three points we can make a big step towards our ambition.
“They really fought for everything, from the start we were the better team and had the better chances.
“Still we made it difficult for ourselves. If Dušan [Tadić] is scoring it’s three-nil and it’s a big difference, but it happened.
“We expect the reaction of Stoke,” Koeman continued. “Very fast it was two-one and after that I was afraid.
“But the boys showed great character to be calm on the pitch and we got the control back in the game.”
Saints may have won more comfortably, had Tadić been awarded a penalty in the second half and Mané not been sent off late on. Koeman felt referee Lee Mason got both decisions wrong.
“I had a chance to see the clip and after that it’s really easy, it’s 100 per cent a penalty,” Koeman affirmed. “Maybe you talk about the red card of Sadio. In my opinion it wasn’t a red card.”