The departures of right back Nathaniel Clyne and defensive midfielder Morgan Schneiderlin in the summer appears to have unsettled a defence that conceded just 33 goals in the league last season -- second best only to champions Chelsea (32).
Defender Maya Yoshida said the Saints must shake themselves out of their early season slumber, especially with a Europa League tie against Danish side Midtjylland coming up midweek.
"Nobody expected it to go like this because it was our opening home game," Yoshida told Southampton's official YouTube channel. "We made completely the same mistake as last Sunday against Newcastle from the set-piece.
"The second goal was a mistake. It was too easy to concede the goals and I think Everton played like we wanted to play. They deserved to win the game."
After two straight seasons of selling their top talent to Premier League rivals, Southampton's early form suggests they will struggle to match last year's seventh-placed finish.
"We have a game in the Europa League on Thursday so it's a very important wake-up alarm for us," added Yoshida.
"We have to concentrate to make a clean sheet. We need to be more compact, more solid physically and tactically we need to be more intelligent. We need to improve a lot of things.
"We will talk about the previous game, with the coaching staff and the team. We have to talk and I hope we can get our confidence back on Thursday."
Koeman, who had surgery to fix his Achilles injury just before the season began, said the changes to his defensive personnel could be behind the lack of cohesion.
Southampton signed central defender Steven Caulker on a season-long loan from relegated Queens Park Rangers late last month after Romanian Florin Gardos went down with a knee injury that could keep him out for seven months.
"Last week we received the second goal at Newcastle out of a corner kick. You can't do those kind of mistakes," Koeman told the club's website.
"We don't have a lot of options defensively at the moment and maybe that makes everyone a little bit more unsure than normal.
"We have to accept defeat, learn from it and keep going."
Southampton travel to Watford on Sunday for their next Premier League fixture.
(Writing by Peter Rutherford; Editing by Greg Stutchbury)
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