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.
“The turning point was not to concede a second goal,” admitted Wenger, whose side were almost caught out as they went in search of the equaliser in the dying stages.
“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 point.
“Maybe we didn’t deserve to win it because of the quality of match Anderlecht played tonight.”
Arsenal sit three points behind group 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 in two weeks.
Besnik Hasi’s side, in contrast, were left deflated after suffering a sixth consecutive home defeat in the group stage, falling moments short of a first victory over English opposition since beating Manchester United 2-1 at the same venue almost exactly 14 years ago.
“My team was almost perfect, except for the last three minutes. We should have won the game, definitely,” said former Albanian international Hasi, who also insisted that Podolski’s decisive goal should have been disallowed for a foul on Steven Defour.
The visitors had been forced to hand a Champions League debut to 22-year-old Argentinian goalkeeper Emiliano Martinez with regular custodian Wojciech Szczesny suspended following his sending-off in the win against Galatasaray three weeks ago and would-be replacement David Ospina out injured.
However, Martinez had almost nothing to do in the first half, as Anderlecht’s limitations were exposed — they defended doggedly and competed well, but failed to produce a single shot on target.
– Lacking in inspiration -Nevertheless, Arsenal were similarly lacking in inspiration, despite the lively running of Danny Welbeck and a few crafty Jack Wilshere passes.
Things improved for Wenger’s men within five minutes of the restart though as Santi Cazorla’s free-kick found Alexis Sanchez unmarked in the area. However, with his back to goal, the Chilean’s header was blocked by goalkeeper Silvio Proto.
Cazorla was then denied by Proto after cutting into the box from the left with 63 minutes played, and that was to prove a costly miss by the Spaniard as Najar put the hosts ahead eight minutes later.
The Honduran winger exchanged passes with Dennis Praet before continuing his run into the box and heading the right-wing cross that followed low into the right-hand corner of Martinez’s net.
It could have got embarrassing after that for Arsenal, with Anthony Vanden Borre hitting the bar and Matias Suarez being denied by his compatriot Martinez moments after coming off the bench.
Martinez also proved his worth with a stop to deny Defour before, from nowhere, Arsenal levelled in the 89th minute.
A Chambers cross from wide on the right travelled all the way to the back post, where left-back Gibbs connected first-time on the volley to leave Proto helpless.
The momentum had now swung their way, and Arsenal silenced the home support when Sanchez’s low ball into the danger area was turned home by Podolski, who wheeled away to celebrate with the visiting fans.