A first-half goal from Raheem Sterling had put the hosts in front, but Nathaniel Clyne hit an emphatic equaliser 10 minutes after the break as Saints enjoyed a good spell with Simon Mignolet thwarting Steven Davis from inside the area.
Despite waves of Southampton pressure, it was Liverpool who claimed the three points as Daniel Sturridge's deft touch took the ball beyond debutant goalkeeper Fraser Forster following Sterling’s header with 11 minutes remaining.
Morgan Schneiderlin then rattled the bar three minutes from time and Shane Long nodded the rebound wide from close range with the best chance of the game as last season's runners-up just about held on.
Koeman told Sky Sports: "It's difficult and I'm very disappointed about the final score.
"We played a great second half and had the best chances. We deserved not to lose today but that's football.
"We had the best chances with Steven Davis and Shane Long and we deserved that goal, but the final score is what counts.
"We had a lot of changes in the team but we brought in quality players. The rumours were that we would be worse than last season but I don't agree with that.
"The performance was very good and it gives the boys a lot of confidence."
Asked about potential further arrivals this summer, Koeman added: "We need one or two players more, a winger. The club is working hard."