The visitors impressed at Anfield and pushed last season's second-place finishers all the way before Daniel Sturridge's winner, with the new manager confident for the season ahead
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, with the visitors enjoying further chances after equalising to take the lead before Sturridge's telling late blow.
"I think we played a great, impressive second half," Koeman told reporters. "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 lots of opportunities to score 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. I don't think we will struggle."