"That's the best present on your birthday, three points and a comfortable win," Koeman said after the game but he did admit that Burnley had given his team a tough examination in the first half.
"The first half was difficult, we didn't play very good," he added. "We lost a little bit of the control in a few moments in the first half, but I was very pleased about the performance in the second half.
"We had total control in the game, good ball possession, good movement and good chances. It was a good win. The most important was the three points and to keep pressing the teams above us in the table and that's important.
"Now it's the international break and then we have Everton away. We have always said game by game and that's the best thing to do until the end of the season."
Southampton clinched the win despite losing goalkeeper Fraser Forster to a knee injury early in the game, with Kelvin Davis coming on as a substitute. Koeman said: "We had to change Fraser, but Kelvin did a great job and that's always good to have an experienced goalkeeper on the bench. He looked very comfortable.
"He had one or two great saves in the first half and that was important for the team. Of course, he likes to play always but he knows his situation behind Fraser. Then if you get your chance you have to be on your level. He was great today and that's all about experience. He was joking after and said maybe they will take him with the national team.
"The news on Fraser is that there is something on his knee. He is now taking a scan and we have to wait for news about that. It's not looking good."
Davis wasn't to be beaten, but Clarets' boss Sean Dyche thought he should have been and also rued the lack of a penalty award when George Boyd went down in a challenge with Jose Fonte, the Southampton captain.
Dyche said: "I doubt they would have expected to go in a goal up at half time because I felt we played ever so well. We came down here, played on the front foot and created four good chances and should have had a penalty.
"That's three we haven't been given in recent weeks and I have no problems with referees, but that was a penalty today. I ask my players not to simulate, but that might have to change because I must be the greenest manager in the world looking at that. Therein lie the margins of the Premier League, unfortunately."
Regarding those chances, he added: "I played with Kelvin many moons ago (at Luton in 1999) and he is a fine keeper, but it was more our poor finishing today. They were a good height for the keeper and we have to learn that at this level, you have to take those chances.
"I was a bit disappointed with that but I rarely question my strikers because I want them to play with a freedom to miss chances, but today I felt they were very good chances that we should have taken.
"The second half was flat until they got the goal and then it's tough for us, but we kept trying to effect the game and that's how we play. We never give up, so we'll dust ourselves down and go again."