The Gunners were stretched in defence before the 3-1 fourth-round home loss after cruciate knee ligament injuries had ruled out Rob Holding and Hector Bellerin for the season.
Arsenal manager Unai Emery also feared that French central defender Laurent Koscielny might have suffered a broken jaw in the United game at The Emirates but it was only bruising.
In an injury update on their website, Arsenal said the 30-year-old Sokratis had injured his left ankle and was "expected to return to full training by the end of February".
Koscielny, 33, suffered bruising and soft tissue swelling to his face and jaw and was "being reviewed on a daily basis".
Arsenal are fifth in the Premier League on 44 points from 23 games, three points behind Chelsea who are in the last Champions League qualifying spot, and host Cardiff City on Tuesday.
(Writing by Ken Ferris; Editing by Clare Fallon)
