Koeman is Everton’s top target to replace fired Roberto Martinez, but the Dutchman will turn down the Toffees to guide Saints in Europe for the second season in a row.
Sadio Mane, Graziano Pelle, Ryan Bertrand and Steven Davis all scored as they did their best to convince their gaffer to sign a new contract at St Mary’s.
But Koeman insists he won’t sign a new deal just because the south coast team are playing in Europe next season.
Koeman, who will be a free agent next summer, said: “Signing a new contract is not depending on European football, it is about the ambition of the squad.
“I like to stay here, I have one year left of contract and from the start I have had an incredible understanding with the fans but it is all about the results.”
Crystal Palace keeper Julian Speroni handed Saints the advantage with a howler, with a weak punch which Mane took advantage of to loop home.
Substitute Pelle headed in a second before Jason Puncheon threatened to break the net with a thunderbolt strike to grab a goal back.