The Greece international, who had played every minute in Arsenal’s first six matches of the season, was replaced by Rob Holding in the 39th minute of the first half after feeling discomfort in his knee.
Goal understands Sokratis picked up the injury during a coming together with Everton winger Theo Walcott and he is now expected to have his injury assessed by doctors on Monday.
Sokratis, 30, became one of Unai Emery’s first signings at Arsenal after arriving from Borussia Dortmund in a deal worth a reported £17.7 million ($23m) in July.
He has been an ever present at the heart of the Arsenal defence since his arrival, having formed a partnership with German centre-back Shkodran Mustafi.
But it’s now feared that the former Dortmund man could be ruled out for the foreseeable future, with Holding on guard to replace him in his absence.
The Gunners face four matches in quick succession before the next international break and a spell on the sidelines for Sokratis would be a blow for Emery, who is still without club captain Laurent Koscielny after he sustained an Achilles injury in May.
Sokratis’ countryman Konstantinos Mavropanos would be another centre-back option for Arsenal but is yet to feature this season because of a groin injury.
Petr Cech, who has been subject to a fair amount of criticism so far this season, was Arsenal's best player in their win over Everton, with his attacking team-mates effusive in their praise of the goalkeeper.
"A big thanks to Petr Cech because he took a lot of balls today and we won our first clean sheet," Alexandre Lacazette, who scored the opener, told Sky Sports.
Pierre-Emerick Aubameyang, who scored the second goal, added: "I think 20 times maybe, Cech kept us in the game."