Koeman's men made it seven points from their first four Premier League games with a dominant victory at St Mary's Stadium.
Striker Graziano Pelle scored a first-half double before midfielders Jack Cork and Morgan Schneiderlin hit home in the second period as Southampton replicated the result from last season's corresponding fixture.
And, while Koeman was not completely satisfied with the display, he was full of praise for a team that many had written off after seeing several key players sold by the club in the transfer window.
"It wasn't perfect because we made some mistakes in the first half. We had a fantastic first goal. We controlled it totally," Koeman said.
"After 30 minutes, we didn't press like we did before. Our goalkeeper Fraser Forster did a great job on two occasions.
"After the third goal, Newcastle went down in a mental way. It was a good performance. Everything is going well."
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