With Ramsey also going off before the final whistle with a groin problem, pressure will mount on Wenger to buy before the transfer window closes.
"We played 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," said Wenger.
"He (Wilshere) looks all right. But Podolski is definitely out for 21 days and Ramsey we have to check how severe his groin is tomorrow (Wednesday)."
Semi-finalists in 2011 and knocked out in the last 16 last season, Schalke also returned to the group stage after getting the better of Greek club PAOK.
PAOK held the upper hand after a 1-1 draw in the first leg in Germany, but lost some of their advantage playing at home behind closed doors.
The German team clinched a 3-2 win on the night to progress 4-3 on aggregate.
Elsewhere, 1986 European champions Steaua Bucharest of Romania returned to the group stage for the first time since 2008-09 after sneaking past Legia Warsaw on away goals.
Swiss champions Basel beat Bulgaria's Ludogorets Razgrad 2-0 for a 6-2 aggregate success, while Austria Vienna finished with a 4-3 aggregate victory against Dinamo Zagreb.
The remaining play-off ties will be played on Wednesday, including AC Milan v PSV Eindhoven, Real Sociedad v Lyon and Celtic v Shakhter Karagandy of Kazakhstan, with the Scots looking to overturn a 2-0 first-leg deficit.
-AAP