Southampton boss Ronald Koeman felt his team deserved a lot more and was unhappy referee Kevin Friend did not give at least one penalty after two challenges on Filip Djuricic.
The Dutchman told Sky Sports: "We know in football it's all about scoring goals. I think we played much better than Liverpool. We had more ball possession, more chances.
"I call it unlucky. And with some decisions of the referee. The second situation is a clear penalty. I'm very disappointed about the result but not about our performance. We deserved much more than we got.
"Maybe the experience of them and the real quality at the front we're a little bit struggling because we scored one goal in the last four games. It makes it a little bit more difficult."
Liverpool boss Brendan Rodgers disagreed with Koeman about the possible penalty and hailed his side's defending after a fifth successive clean sheet away from home in the Premier League.
He said: "The first one, the player's gone over far too easy. With the crowd up, of course that puts the referee under pressure but I thought Kevin was excellent. All his decisions were correct.
"We defended magnificently. We knew we needed to improve. At the beginning of the season we couldn't get the right balance.
"Now there is and there's great confidence in the team. Our game still could be better but the result was important and everyone's contributed."
Rodgers was particularly impressed with the resilience his side showed only three days after their Europa League win over Besiktas.
He said: "We knew it would be a huge win if we could do it and it was. I thought we were really well organised and we controlled the game as it went on.
"Our passing could have been better but they were difficult conditions and it's an outstanding win. It's difficult to come here at any time and get a result but to do it on the back of a European result is brilliant for the players."