As well as having to sit out the Doncaster game, he will miss the Premier League fixtures with West Brom and Brighton, and the Europa League tie at BATE Borisov of Belarus.
Playmaker Mesut Ozil, midfielder Francis Coquelin and full-back Mathieu Debuchy will also be absent on Wednesday.
The England star who started the game was forced off after 73 minutes at Stamford Bridge. "He has a little inflammation of his knee".
But as a result of Arsenal's improved display, Wilkins reckons manager Arsene Wenger should consider dropping Ozil when they play away from home as he is a liability to the team.
The Gunners haven't lost to lower league opposition at home in the League Cup since November 1983 against Walsall, winning 25 of their last 28 contests, including the last seven in a row.
The Gunners beat the Germans 3-1 in that controversial, late-starting Europa League game game last week, with France striker Olivier Giroud starting.