The Wales international sustained the problem in the first half of the 1-1 Premier League draw at the Emirates Stadium.
And Arsenal boss Arsene Wenger confirmed that Ramsey will be missing for around a month in a press conference ahead of his team's UEFA Champions League clash with Galatasaray.
"To be generous with time, four weeks," Wenger said on Tuesday. "If all goes well, three."
The news means Ramsey is certain to miss Sunday's meeting with Premier League leaders Chelsea at Stamford Bridge.
It comes as a huge blow to Wenger, who saw Ramsey miss a large chunk of the previous campaign with a thigh complaint.
Wenger also declared that Mikel Arteta would be out for a similar period of time with a calf problem although, on a more positive note for the Frenchman, Jack Wilshere is fit despite injuring his ankle against Spurs.
"He [Wilshere] is in the squad [for Galatasaray]," Wenger added. "Will I start him or not? I don't know, we have to be cautious. I left [Abou] Diaby out as he isn't ready."
