The France international suffered a small fracture near his ankle in the final seconds of Arsenal's 2-2 draw against Everton on Saturday after the ball smashed into the sole of his boot, forcing his foot towards his shin.
Following Arsenal's 1-0 win over Besiktas at the Emirates Stadium on Wednesday, which sees them qualify for the Champions League groups stages, Wenger told Sky Sports 5: "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."
Despite losing Giroud for up to four months, Wenger has ruled out panic-buying a replacement, instead choosing to hail Arsenal's quality up front.
"We will try to look around but we want quality and we have quality," the Frenchman said.
"We have fortunately (Theo) Walcott coming back soon, I believe (Yaya) Sanogo will have an impact. We have (Lukas) Podolski, and (Joel) Campbell who can play centre forward so to find players that are better than the players we have will be difficult.
"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."
Wenger also confirmed he sees summer signing Alexis Sanchez starting up front.
"I bought Samchez to be a striker and I think Walcott can be a good striker as well, Sanogo can be a good striker," Wenger added.
"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."
Despite the Giroud news, Wenger could at least celebrate Arsenal's progression to the group stages of the Champions League on a night of high anxiety at the Emirates Stadium that saw Sanchez open his Gunners' account.
The Chile striker operated as a lone striker and claimed the vital goal which enabled Arsenal to progress into the group stages of European football's premier club competition for a 17th successive season on the stroke of half-time.
"It is unthinkable until it happens and then everyone thinks 'why did it happen'," Wenger said on the thought of his side not reaching the group stages.
"It is a weight off our shoulders now because we can focus on the Premier League, wait for the draw. We feel we have done the job and we have a clear conscience."
Arsenal had to finish the match with 10 men after Mathieu Debuchy was sent off for collecting a second yellow card.
Wenger added: "I was very worried because you never know as we have some players on the edge physically.
"I thought it was very harsh because Debuchy played the ball and maybe with the anger from our side on the bench we saw he played the ball. The referee was on the other side."
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