The Spain midfielder hobbled off in the early stages of the 1-0 Premier League win over Stoke City on Wednesday.
"It is a very small injury but certainly Cesc will be out for Sunday," Arsenal manager Arsene Wenger said on the club's website. "For how long (beyond that) I don't know but for Sunday he will be short.
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"He is (disappointed). We all feel sorry and sad for him. The only way we help him now is to win the Carling (League) Cup as he contributed a lot in this competition."
Wenger would not be drawn on the chances of Fabregas being fit for the second leg of the Champions League last 16 tie against Barcelona at the Nou Camp on March 8.
Arsenal lead 2-1 from the first leg at Emirates Stadium.
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The London club's England winger Theo Walcott will also miss the League Cup final after being carried off on a stretcher against Stoke although his ankle problem may not be too serious.
"He has a classic ankle sprain," said Wenger.
"We don't think there is any more damage to it. But it is still a sprain. We are sad for him too. He is out for Sunday and maybe one or two more weeks."
However, striker Robin van Persie (hamstring), central defender Laurent Koscielny (back) and midfielder Abou Diaby (calf), are available for Sunday's showpiece after missing the home win over Stoke.
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