However, Daniel Sturridge scored late on for the home side to secure them all the points.
"It's difficult and I'm very disappointed about the final score," he told BBC Sport.
"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. We need one or two players more, a winger. The club is working hard."
Koeman took charge of the Saints earlier this summer following Mauricio Pochettino's switch to Tottenham Hotspur.