Southampton ensured a miserable return to St Mary's for the manager who left them in the summer, outplaying them to such an extent they deserved a wider margin of victory than that provided by Charlie Austin's first-minute goal.
Everton were as flat as Southampton were in Thursday's 1-0 Europa League defeat by Sparta Prague
That they are seventh in the Premier League - the highest outside of those expected to compete for the title - leaves them three places above their hosts but a two-point advantage will diminish if they cannot arrest their poor run.
The visitors lost possession almost immediately before Southampton's impressive one-touch football led to debutant Josh Sims teeing up Austin to head in after 41 seconds
They thereafter struggled to impose themselves, highlighting the extent of the challenge Koeman accepted when leaving the south coast.
"I'm not worried about results, I'm worried about what I see on the pitch," said the Dutchman, who was widely considered to be progressing his career when leaving Southampton for Everton.
"How we start the game, have difficulties to be organised defensively and stop the opponents
I'm worried because the wake-up call, how we played in first half, to concede a goal after one minute: that cannot happen.
You start a game to control a game and we didn't; we had bad defending positions for the cross
That (goal) made it difficult and a different game.
"The whole first half was not good
(We had) problems with (Southampton's) strikers, and that's strange because we know how they play
We need to be sharp, react fast to second balls
But we had some good chances in the first half; the shot from (Idrissa) Gueye needs to be on target."
Asked of Ross Barkley's and Romelu Lukaku's performances, Koeman answered they were "not good enough", before also revealing James McCarthy's absence owed to a precautionary measure taken against a hamstring injury.
The afternoon was significantly more positive for Koeman's successor Claude Puel, even if they were again reliant on Austin's sixth league goal - from Southampton's total of 13 - of the season securing all three points.
His team selection was rewarded with a fine display from the 19-year-old Sims - Koeman told the BBC: "I wasn't aware of Josh Sims when I was here and he is not my problem; I work for Everton and not Southampton" - and Puel praised his "very good" performance.
"For a first game in the Premier League it wasn't bad," the Frenchman said as he smiled
"It was a very good game for him
He made a good assist (for Austin's goal).
"It was not a risk to start him and I could see really good things in training and Under-23 games; he has made good progress this season
It is just one game but for a start it is good to see from him.
"The club have a good philosophy for young players
For me it is important to continue this work and philosophy
It is important for Southampton to develop young players and I respect this."
Source : PA