Daniel Sturridge poked home with 11 minutes to play to seal an unconvincing win for Liverpool as both sides opened their Premier League campaigns.
But the nature of Southampton's performance – particularly after the break – pleased Koeman, who has seen his team ravaged by the sale of several members of his squad in the close-season, most notably the departure of a trio of players to Liverpool.
Nathaniel Clyne cancelled out Raheem Sterling's first-half effort for the visitors and Southampton had further chances to take the lead before Sturridge's telling late blow.
Koeman was buoyant afterwards, though, and has urged his players to believe in themselves.
"I think we played a great, impressive second half," Koeman said.
"After the [Clyne goal to make it] 1-1, I think there was one team on the pitch playing and that was Southampton.
"We did everything well, minus to score the second one. We create two great possibilities, opportunities, but the quality of Liverpool – they don't need a lot of opportunities to score two goals and that made the difference.
"The way how we played, I think that brings us a lot of confidence. And if we play like [that], the points will come.
"We have to believe in our qualities. That was one of the messages that we gave the boys before the match.
"Little by little, we did it. We played a great, impressive second half and that's the way we like to play.
"If we play like that, with that team spirit, I don't think we will struggle [for] the whole season."
One area that Southampton can improve on, though, is their defensive shape, something Koeman intends to target in the coming weeks.
"How we did the pressing, the high pressing, in the second half, it was a little bit different than in the first half," he added.
"I think in the first half we lost a little bit of attention in the transition to attack of Liverpool, because they got two chances about that, one was the goal.
"It was not a good defensive organisation at that moment [when Sterling scored].
"[But] we played well, we deserved a better result and that gives us a lot of confidence."
