Schneiderlin stay crucial - Koeman
Schneiderlin was highly sought after by former coach Mauricio Pochettino, who was determined to take the France international to Premier League rivals Tottenham during the transfer window.
The 24-year-old, who has been at St Mary's since arriving from Strasbourg in 2008, also criticised Southampton for blocking his move away.
But despite Schneiderlin's discontent, Southampton refused to part ways after losing captain Adam Lallana, Luke Shaw, Dejan Lovren, Calum Chambers and Rickie Lambert before the start of the season.
And Koeman's stubbornness appears to have worked, with Schneiderlin now back to his best, having scored two goals in the first three league matches.
"I hope keeping Morgan is our best piece of business because he is a very important player for our team. He is playing in an important position," Koeman told reporters.
"We had a difficult two weeks but he did everything in a good, professional way and the best way to answer is on the pitch.
"It was an important decision to keep him, not only for the quality of the player but for the spirit and ambition of the club and for the rest of the team.
"Morgan was disappointed at first but that is a normal reaction and after that his behaviour was professional.
"I had several conversations with Morgan but finally it was the player who made the right decision. From what I have seen during the last three weeks he is a fantastic player."
Tottenham are expected to renew their interest in Schneiderlin in January, though Koeman refused to speculate the player's long-term future with Saturday's hosting of Newcastle United on the forefront of his mind.
"I don't know about that, we will just have to wait and see," he said.
"I'm not thinking about January at the moment, I'm just preparing our team for Saturday's Newcastle match.
"That's the most important thing at this moment."