Koeman – who managed the Dutch club earlier in his career – named a strong side for Thursday night’s game, and it paid off as Graziano Pelle, Dusan Tadic and Shane Long scored to put Saints 3-0 up in the tie.
It was a strong showing for the south coast team resting Jay Rodriguez, but there was one big concern after new arrival Jordy Clasie was withdrawn on his debut with what looked like a hamstring injury.
“Clasie did well,” Koeman said. “He had a good help of (Victor) Wanyama, who was outstanding and that makes it more easy for Clasie. Pity he had some hamstring problems and we have to wait to know what happened.”
The Saints boss though was delighted with the scoreline and added: “It’s never over in football, but of course the final result is a very good one.
“In European football, to win the first game at home 3-0 is a great result. It’s a very good feeling; I think we had a great atmosphere in the stadium.
“The fans were looking forward to the game. After the first 15 minutes we controlled the game well and we had a great first goal. Good movement, good passing and a great finish by Pelle, and, of course, the second – the penalty – makes it more easy for the second half.”
Southampton scored once in the second half after a bad mistake by Chelsea loanee Lewis Baker, but Koeman was disappointed that his side took their foot off the gas.
“I wasn’t happy about the second half because I think we played too much defending the result, and were lucky with the mistake of Vitesse to score the third one,” he concluded.
Saints now travel to Holland next week for the return leg, where they can book their place in the play-off round of the Europa League.