Sadio Mane set the visitors on their way at Selhurst Park with a superb goal after 17 minutes, and then second-half strikes from Ryan Bertrand and Toby Alderweireld wrapped up the win, despite Scott Dann's late consolation.
It marked the first time Southampton have secured back-to-back wins since November and Koeman was left in no doubt that his side deserved the three points.
He told the club's official YouTube channel: "I think we had a good start, we controlled the game and had some good opportunities.
"Maybe the last 15 minutes before half-time we had a little bit more problems to be the dominant team.
"But we defended well and we looked solid today. It was a good performance. It was important [the fast start to the second half].
"We showed our spirit to try to score the second one and not to defend the one-nil. That was the difference between us and our opponent today."
The performance of Mane was a particular highlight for the visitors, with the Senegal international scoring the first and setting up the second during a largely comfortable afternoon.
"I'm very happy for him [Mane]. He played a little bit of a different position but he knows that position, he’s not fazed by that," added Koeman.
"It gave him more freedom in his running and his movements and he scored a great goal. It was a great pass from Prowsey [James Ward-Prowse], that was important for him and the team.
"We know we have to play two games in 48 hours, we need all of the players and that was part of the decision to bring him in, but he did it well."
