Aaron Ramsey scored twice for the London side as they reached the group stages for the 16th successive year with a 5-0 aggregate victory, but along with fellow midfielder Jack Wilshere he also suffered an injury scare late in the match.
"That's the negative of tonight," Wenger told reporters. "We paid a heavy price on the injury front because we lost Podolski. I don't know how bad Ramsey is. I don't know how Wilshere responds to an ankle problem. We have to check that.
"The two tackles on Wilshere were disappointing. I cannot change it now. He looks alright. Podolski is definitely out for 21 days and Ramsey we have to check how severe his groin is tomorrow."
Wenger said he was hopeful that Wilshere and Wales international Ramsey would be fit for the north London derby with Tottenham Hotspur on Sunday, but would not say if it would hasten his efforts to bring in new players before the end of the transfer window.
Arsenal have already lost Alex Oxlade-Chamberlain for up to six weeks with a knee injury, while Mikel Arteta has a thigh problem.
"We have to see how bad the injuries are. I cannot tell you more than we are on the market. Everybody knows. The whole world knows that now. I cannot be more creative than that in the press conferences because all has been said and nothing has been concluded yet so I can understand your questions but I cannot give you more answers at the moment."
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- Jack Wilshere