Giroud saved his side from a defeat,levelling the tie in their 2-2 draw against Everton last Saturday but he ended getting injured.
Speaking on Sky Sports after Arsenal made it through to the Champions League group stage with victory over Besiktas, Arsene Wenger confirmed his compatriot is facing a long absence.
"The news is that he had surgery today and the surgery went well but he will be out - maybe to be competitive at the top, top level - until the end of December, beginning of January."
Wenger used Alexis Sanchez at centre-forward against Besiktas, and the £35m summer signing scored the only goal to send his new side through.
Yaya Sanogo, Lukas Podolski and Joel Campbell are other options, while Theo Walcott is close to returning from a long-term injury of his own.
Arsenal manager revealed he wasn't sure he whether he would bid for another striker before the September 1 transfer window deadline.
"We will try to look around but we want quality and we have quality," he said. "I bought Sanchez to be a striker and I think Walcott can be a good striker as well.
"Sanogo can be a good striker, Podolski can play there and Campbell if you ask what he is, he's a striker. They are players of quality and we will see.
"We will concede in England that every time you have a problem you can't just buy. After the first of September you still have to live with it."
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