An Andy Najar header in the 71st minute looked set to give the Belgian champions a rare European victory at the Constant Vanden Stock Stadium until Gibbs volleyed in a Calum Chambers cross with a minute left.
Substitute Podolski then struck a minute into injury time to seal all three points, to the great relief of Gunners' manager Arsene Wenger on his 65th birthday.
After a run of just two wins in nine games in all competitions, it looked set to be another evening to forget for Wenger, but the Frenchman can breathe a sigh of relief thanks to a result that leaves the Premier League club in a strong position in Group D.
"They had big chances that they didn't take, and our experience made the difference. Unexpectedly, but as long as we didn't concede the second goal we had a chance at least to take a pont.
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"Maybe we didn't deserve to win it because of the quality of match Anderlecht played tonight."
Arsenal sit three points behind leaders Borussia Dortmund, who were convincing 4-0 winners away to Galatasaray in the night's other game, but are five clear of the sides below them and well on course for a place in the last 16 with the reverse fixture with Anderlecht to come next.