Despite having their preparations for the game disrupted by a late arrival at Selhurst Park due to heavy traffic, the Saints were full value for the three points thanks to goals from Saido Mane, Ryan Bertrand and Toby Alderweireld.
Concerns had been raised when Southampton endured a run of five straight defeats in all competitions, which included a humiliating Capital One Cup quarter-final defeat to League One Sheffield United.
But they got back to winning ways against Everton last weekend, and they were rarely troubled by Palace from the moment the classy Mane fired them into a 17th-minute lead.
“We had some weeks where we didn’t have all the players available and that is sometimes difficult because we don’t have the quality in numbers,” Koeman said.
“We had (Dusan) Tadic, (Victor) Wanyama and (Shane) Long on the bench today and it means we can change it. It is important we show the spirit and belief we showed – we need everybody.”
Next up for Saints is a home clash with leaders Chelsea on Sunday, and Koeman hinted he is likely to ring the changes to spread the workload over the busy festive season.
“We need all our players, you can’t play with 11 players two games in 48 hours. You have to make some changes,” he added.
“We have those kind of players now that I can change the team. We kept the quality and the spirit, and it was a very solid win.
???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.
???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.???