A magnificent 30-yard strike from Philippe Coutinho gave the visitors an early lead, while Filip Djuricic had a pair of strong penalty appeals rejected on his first Southampton start.
Liverpool withstood plenty of pressure before Raheem Sterling made the points safe in the 73rd minute and Koeman was happy with the manner of his side's performance.
He told Sky Sports: "We know in football, it's all about scoring goals. We played much better than Liverpool, we had more possession, more chances.
"I call it unlucky in the decisions of the referee. The second situation is a clear penalty to Filip Djuricic.
"Two-nil down, I'm very disappointed about the result, but not our performance.
"With the quality up front maybe we are struggling a bit in last few games because in the last four games we score one goal."
Koeman also praised Coutinho's moment of inspiration as Southampton goalkeeper Fraser Forster was left with little chance as the ball clattered home via his crossbar.
"It is a great goal, the keeper can't do anything about it," Koeman added.
"Great goal but we had a great start in the game and we didn't deserve to be 1-0 down.
"We have more ball domination, we create four or five good positions to score and the referee didn't whistle for a clear penalty.
"Sometimes you need that luck on your side and in the last few games we don't have that."
