Wilfried Bony handed the Swans an 11th-minute lead, powering home Neil Taylor's cross.
But the Gunners hit back with two second-half goals in the space of 66 seconds.
Substitute Lukas Podolski bagged his first Premier League goal of 2014 before setting up Olivier Giroud for his 19th goal of the season.
However, the visitors picked up a share of the spoils when Leon Britton's shot was partially blocked by Per Mertesacker, then ricocheted off goalkeeper Wojciech Szczesny and back onto Mathieu Flamini before rolling into the net for an unfortunate own goal.
Arsene Wenger's side will now be looking over their shoulder with Tottenham and Everton breathing down their neck for fourth position.
Alex 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 Taylor and Bony simply had too much power for Vermaelen as he out-jumped the Belgian to head past 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 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, none more so than on the final whistle when Jonathan De Guzman looked to be in behind the defence, only for referee Lee Probert to end the match.
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.