Having trailed to an 11th minute header from Wilfried Bony, it was not until the introduction of substitute Lukas Podolski just before the hour that Arsenal finally sparked into life with two goals in the space of 60 seconds which looked to have won the match.
First, Kieran Gibbs - whose red card at Chelsea in a case of mistaken identity was overturned on Monday night - darted down the left and supplied a low cross which was bundled home by Podolski in the 73rd minute.
The German then turned provider as he charged clear of the Swansea defence to send in a low cross which Olivier Giroud smashed past Michel Vorm at the near post.
However, Garry Monk's side - fighting against being dragged into the relegation battle - had defended resolutely and were themselves dangerous on the counter-attack, eventually forcing a late equaliser.
Following a disputed free-kick, the ball broke to Leon Britton at the edge of the Arsenal penalty area.
As the midfielder pressed forwards, riding a couple of challenges, he stabbed the ball goalwards
It was partially blocked by Per Mertesacker, then ricocheted off goalkeeper Wojciech Szczesny and back onto Flamini before rolling into the net.
Wenger was left less than impressed, and will be in no doubt of the challenges which now lie ahead in a campaign which has promised so much - and continues with the visit of Manchester City on Saturday.
Arsenal had both midfielder Oxlade-Chamberlain and left-back Gibbs in the side after successful disciplinary appeals from the fallout of Saturday's fiasco at Stamford Bridge
The Football Association overturned the mistaken-identity dismissal of Gibbs by referee Andre Marriner for Oxlade-Chamberlain's handball in the six-yard box, which was then downgraded to a yellow on review as it was decided the initial shot had been going just wide.
Thomas Vermaelen started at centre-back with Laurent Koscielny out because of a calf problem, while defensive midfielder Flamini came in to shore up the side - as many felt the Frenchman should have done at the weekend.
Swansea are in danger of being sucked into the relegation dogfight after no wins in the last six league matches, but recalled Spanish forward Michu as one of four changes by Monk.
Oxlade-Chamberlain almost found himself in the headlines again on five minutes when his rasping 20-yard drive was palmed away by Michel Vorm.
It was the visitors, though, who took the lead on 11 minutes
The ball was swung over from the left by Neil Taylor and Bony simply had too much power for Vermaelen as he out-jumped the Belgian to head past Wojciech Szczesny
Arsenal looked for an immediate response as Tomas Rosicky charged into the Swansea penalty area, but was crowded out before Mertesacker's flick at the near post from a corner went into the side netting.
Swansea, though, continued to look dangerous on the counter attack as, just like against Chelsea, Arsenal were caught out pressing a high line.
When Cazorla did squirm free to make space for a shot from the edge of the crowded Swansea penalty area, Vorm saved smartly.
Arsenal started the second half with some purpose, but were again found wanting of a telling pass in the final third.
Wenger had seen enough and made a change on 57 minutes as Podolski replaced Oxlade-Chamberlain.
It could have got worse for Arsenal had Michu not sent a free header over the crossbar after another floated ball into the penalty area.
Arsenal suddenly sparked into life to turn the match around with two goals in a minute.
Gibbs got away down the left and to the goal-line before cutting it back into the six-yard box, where Podolski arrived on queue to hook his shot over the keeper.
The German then turned provider as his powerful run and cross was finished off by Giroud at the near post.
Despite Arsenal turning the match around, there were still plenty of nervous moments in the closing stages.
And the home supporters were left in stunned silence at the dramatic finale as Swansea players ran over to the jubilant travelling fans, with Flamini and the rest of Wenger's men left to curse their luck once again.
Source : PA